Interactive Manual › Chapter 10

Chapter 10: Matrices

Define, edit, display and calculate with matrices [A] to [J]: the MATRX menus and matrix editor, the MTRX shortcut, matrix arithmetic, and the MATRX MATH functions such as det(, rref(, augment( and the row operations.

Getting Started: Using the MTRX Shortcut Menup. 143

The MTRX shortcut (ALPHAF3, the [F3] function) lets you type a small matrix straight into a calculation on the home screen or in the Y= editor, without visiting the matrix editor first. It is available in MathPrint mode.

Note The quick matrix editor starts every cell at 0. If you want to type a fraction into a cell, delete the pre-filled zero first.

Tutorial

Add two matrices with the MTRX shortcut and store the sum in [C]

You will add [[2,3][5,8]] and [[4,3][2,1]] and store the answer in matrix [C].

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the quick matrix editor. The default size is 2 rows by 2 columns.
ALPHAF3
Move down to highlight OK, then press ENTER to insert a 2×2 matrix.
▼︎▼︎ENTER
Fill in the first matrix, using ▶︎ to move from cell to cell (row by row).
2▶︎3▶︎5▶︎8▶︎
Press +, open a second 2×2 matrix the same way, and fill it in. Press ENTER to calculate.
+ALPHAF3▼︎▼︎ENTER4▶︎3▶︎2▶︎1▶︎ENTER
Store the result in [C]: press STO►, then open the MATRX NAMES menu and choose 3:[C], then ENTER.
STO►2ndMATRIX3ENTER

After the calculation is stored, [C] is a 2×2 matrix. You can see it in the matrix editor (2ndMATRIX, then ▶︎ ▶︎ for the EDIT tab) and edit it there like any other matrix.

Getting Started: Systems of Linear Equationsp. 144

You can solve a system of linear equations by typing its coefficients into a matrix and asking the calculator for the reduced row-echelon form with rref(. The example below solves X + 2Y + 3Z = 3 and 2X + 3Y + 4Z = 3.

Each equation becomes one row: the coefficients of X, Y and Z followed by the constant on the right-hand side. That gives a 2×4 matrix [[1,2,3,3][2,3,4,3]].

Tutorial

Solve a system of two equations with rref(

You will enter the coefficients in matrix [A] and use rref( to read off the solution.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the MATRX menu and move to the EDIT tab (press ▶︎ twice).
2ndMATRIX▶︎▶︎
Choose 1:[A] to edit matrix [A].
1
The cursor is on the row dimension. Type 2 and press ENTER, then type 4 and press ENTER to make a 2×4 matrix. The cursor now sits on the first element.
2ENTER4ENTER
Enter the first element. The cursor moves to the next column of the first row.
1ENTER
Finish the first row (X + 2Y + 3Z = 3): enter 2, 3 and 3.
2ENTER3ENTER3ENTER
Enter the second row (2X + 3Y + 4Z = 3): enter 2, 3, 4 and 3.
2ENTER3ENTER4ENTER3ENTER
Return to the home screen. Press CLEAR to clear the screen if it has old work on it.
2ndQUITCLEAR
Open the MATRX menu on the MATH tab (press ▶︎ once). Press ▲︎ to wrap around to the end of the list, then pick B:rref(.
2ndMATRIX▶︎▲︎ALPHAB
Paste [A] from the MATRX NAMES menu, close the parenthesis and calculate. The reduced row-echelon form appears and is stored in Ans.
2ndMATRIX1)ENTER
Read the result row by row. Row 1 says X − Z = −3, so X = −3 + Z. Row 2 says Y + 2Z = 3, so Y = 3 − 2Z. Z can be any value, so this system has infinitely many solutions.

Defining a Matrixp. 145

Everything about creating and changing a matrix starts here: what a matrix is on this calculator, how to choose which one to work on, and how to set its size.

What Is a Matrix?p. 145

A matrix is a rectangular grid of numbers arranged in rows and columns. The calculator has 10 matrix variables, [A] through [J].

You can display, define or edit a matrix in the matrix editor (2ndMATRIX, then the EDIT tab).
You can also define a small matrix with the MTRX shortcut (ALPHAF3), or type one directly into an expression.
A matrix can have up to 99 rows and 99 columns, as far as free memory allows.
Only real numbers can be stored in a matrix. Fractions are stored as real numbers, so you can use them.

Selecting a Matrixp. 145

Before you can define or view a matrix in the editor you must pick its name. Press 2ndMATRIX to open the MATRX menu, then press ▶︎ twice to reach the EDIT tab. Any matrix that already exists shows its dimensions next to its name.

The NAMES tab is the one that opens first. It is used to put a matrix name into a calculation, while the EDIT tab opens the matrix editor. The middle tab, MATH, holds the matrix functions described in Using the MATRX MATH Operations.

Tutorial

Open a matrix in the editor

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press the MATRX key, then move right twice to the EDIT tab.
2ndMATRIX▶︎▶︎
Choose 2:[B]. A new matrix starts out 1×1 and filled with 0.
2

Accepting or Changing Matrix Dimensionsp. 146

The matrix size (rows × columns) is shown on the top line of the editor. A brand-new matrix is 1×1. Each time you open a matrix you must accept or change its dimensions before you reach the elements.

The cursor starts on the row dimension. To keep it, press ENTER. To change it, type the number of rows (up to 99) and press ENTER.
The cursor then moves to the column dimension. Accept or change it in the same way.
When you press ENTER on the column dimension the cursor jumps to the first matrix element.

Tip Changing the size keeps the elements that still fit and fills any new cells with 0. Elements outside the new size are discarded.

Tutorial

Make [B] a 3×2 matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the EDIT tab of the MATRX menu and select [B].
2ndMATRIX▶︎▶︎2
With the row dimension highlighted, type 3 and press ENTER.
3ENTER
Now the column dimension is highlighted. Type 2 and press ENTER. The cursor lands on element (1,1).
2ENTER

Viewing and Editing Matrix Elementsp. 146

Once the dimensions are settled, the matrix editor shows the grid and lets you change any element. The editor has two modes, called viewing context and editing context, and the cursor keys behave differently in each.

Displaying Matrix Elementsp. 146

In a new matrix every element is 0. The middle of the editor shows up to seven rows and three columns at a time, with long numbers abbreviated to fit. The cursor is a rectangle on the current element, and its full value is shown on the bottom line.

When the matrix is bigger than the window, an ellipsis (...) in the leftmost or rightmost column means there are more columns in that direction, and the arrows # and $ in the right column mean there are more rows above or below. The example in the guidebook is an 8×4 matrix.

MATRIX[A]  2×4
[1  2  3 ...
[2  3  4 ...




2,3=4

Editor layout (illustrative): dimensions on top, elements in the middle, the full value of the current element at the bottom.

Deleting a Matrixp. 146

To remove a matrix from memory, use the delete screen of the MEMORY MANAGEMENT menu (2ndMEM, then 2:Mem Mgmt/Del...). It is described in Chapter 18. Setting a matrix back to 1×1 in the editor does not delete it.

Viewing a Matrixp. 146

When you first enter the elements of a matrix you are in viewing context. Here the cursor keys hop quickly from element to element, and the full value of the highlighted element appears on the bottom line. Select the matrix from the MATRX EDIT menu and accept or enter its dimensions to get here.

Key	What it does in viewing context
`◀︎` or `▶︎`	Moves the cursor along the current row
`▼︎` or `▲︎`	Moves the cursor along the current column. From the top row, `▲︎` goes to the column dimension, and from the column dimension `▲︎` goes to the row dimension
`ENTER`	Switches to editing context and activates the edit cursor on the bottom line
`CLEAR`	Switches to editing context and clears the value on the bottom line
Any entry character	Switches to editing context, clears the bottom line and puts the typed character there
`2ndINS`	Does nothing
`DEL`	Does nothing

Editing a Matrix Elementp. 147

To change one element:

Select the matrix from the MATRX EDIT menu and accept or enter its dimensions.
Use ◀︎, ▶︎, ▲︎ and ▼︎ to move the cursor to the element.
Switch to editing context by pressing ENTER, CLEAR or any entry key.
Change the value on the bottom line with the editing keys below. You may type an expression; it is evaluated when you leave editing context.
Press ENTER, ▲︎ or ▼︎ to store the value and move to another element.

Tip If you make a mistake, press CLEAR then ENTER to put the element back to the value it had before.

Editing-context keys
Key	What it does in editing context
`◀︎` or `▶︎`	Moves the edit cursor within the value
`▼︎` or `▲︎`	Stores the value from the edit line in the element, switches back to viewing context and moves the cursor within the column
`ENTER`	Stores the value, switches back to viewing context and moves the cursor to the next element of the row
`CLEAR`	Clears the value on the bottom line
Any entry character	Types the character at the edit cursor
`2ndINS`	Turns on the insert cursor
`DEL`	Deletes the character under the edit cursor

Tutorial

Change one element of a matrix

Matrix [A] is [[1,2,3][4,5,6]]. You will change the 5 to 9, then type an expression in another cell.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open [A] in the matrix editor and accept both dimensions with ENTER ENTER. The cursor is on element (1,1).
2ndMATRIX▶︎▶︎1ENTERENTER
Move the cursor to row 2, column 2: press ▼︎ then ▶︎.
▼︎▶︎
Type 9 (this switches to editing context and replaces the old value) and press ENTER to store it.
9ENTER
The cursor moved on to element (2,3). Type an expression, 2 × 4, and press ENTER. It is evaluated when stored.
2×4ENTER

Tutorial

Undo a mistaken edit with CLEAR ENTER

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open [A] and accept the dimensions.
2ndMATRIX▶︎▶︎1ENTERENTER
Start typing a wrong value, 77, without storing it.
77
Press CLEAR then ENTER. The original value of the element is kept.
CLEARENTER

Using Matrices with Expressionsp. 148

To use a matrix in an expression you can do any of these:

Paste its name from the MATRX NAMES menu (2ndMATRIX, then the number of the matrix).
Recall its contents into the expression with 2ndRCL (RCL, see Chapter 1).
Type the matrix directly into the expression, as described below.

Entering a Matrix in an Expressionp. 148

Besides the matrix editor, you can type a matrix directly on the home screen or inside any expression. Square brackets are above the multiplication and subtraction keys: 2nd[ types [ and 2nd] types ].

Press 2nd[ to start the matrix.
Press 2nd[ again to start a row.
Type a value (an expression is fine) for each element in the row, separating them with ,.
Press 2nd] to end the row.
Repeat steps 2 to 4 for each remaining row.
Press 2nd] to end the matrix.

When you press ENTER the matrix is displayed in the form [[1,2,3][4,5,6]] and any expressions inside are evaluated.

Note The commas you type between elements are not shown in the answer. Closing brackets are required when you type a matrix on the home screen or in an expression. A matrix typed this way is not stored automatically (a matrix defined in the matrix editor is), but you can store it with STO►.

Tutorial

Type a 2×3 matrix on the home screen and store it

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Begin the matrix and its first row, then type the three elements separated by commas.
2nd[2nd[1,2,3
Close the first row, open the second one, and type its elements.
2nd]2nd[4,5,6
Close the second row and the matrix, then press ENTER to evaluate it.
2nd]2nd]ENTER
It is only the answer so far, not a stored matrix. Store it in [B]: press STO► and paste [B] from the NAMES menu, then ENTER.
STO►2ndMATRIX2ENTER

Tip In MathPrint mode you can also build a matrix with the MTRX shortcut. Press ALPHAF3, choose the number of rows and columns in the dialog, select OK and press ENTER. Then type the elements, pressing ▶︎ to move from cell to cell, and ENTER to calculate.

Displaying and Copying Matricesp. 149

This section covers showing a matrix on the home screen, copying one matrix into another, and reading or changing a single element.

Displaying a Matrixp. 149

To show a matrix on the home screen, paste its name from the MATRX NAMES menu and press ENTER. Matrices bigger than the screen are marked differently in the two display modes:

Mode	Sign that there is more to see
MathPrint	An arrow at the left or right means more columns; an arrow at the top or bottom means more rows
Classic	An ellipsis (...) in the left or right column means more columns; # or $ in the right column means more rows

In either mode, use ▶︎, ◀︎, ▼︎ and ▲︎ to scroll the matrix once it is displayed, including after a calculation. If it will not scroll, press ▲︎ ENTER ENTER to repeat the calculation and try again.

Note You cannot copy a matrix out of the history. Matrix calculations are also not carried over when you switch between MathPrint and Classic mode.

Tutorial

Display a matrix on the home screen

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Paste the name [A] from the MATRX NAMES menu.
2ndMATRIX1
Press ENTER to display the matrix.
ENTER

Copying One Matrix to Anotherp. 150

Press 2ndMATRIX to open the MATRX NAMES menu.
Select the name of the matrix you want to copy.
Press STO►.
Press 2ndMATRIX again and select the name of the matrix to copy into.
Press ENTER.

Tutorial

Copy [A] to [B]

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Paste [A], then press STO►.
2ndMATRIX1STO►
Paste [B] as the destination and press ENTER.
2ndMATRIX2ENTER

Accessing a Matrix Elementp. 150

On the home screen or in a program you can store a value to a single matrix element, or recall one, by giving the row and column in parentheses after the name. The element must lie inside the matrix's current size.

[matrix](row,column)

[matrix](row,column)value→[matrix](row,column)

Refers to one element of a matrix: row first, then column. Use it to read the element in an expression or, with STO►, to change it.

Keys: 2ndMATRIX1(1,2)

[A](2,1) returns the element in row 2, column 1
7→[A](1,2) stores 7 in row 1, column 2

Tutorial

Store and recall a matrix element

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Store 7 in row 1, column 2 of [A].
7STO►2ndMATRIX1(1,2)ENTER
Recall the element in row 2, column 1.
2ndMATRIX1(2,1)ENTER

Using Math Functions with Matricesp. 151

Many of the keypad functions and the functions in the MATH, MATH NUM and MATH TEST menus accept matrices, as long as the sizes are suitable. Each one builds a new matrix and leaves the original unchanged.

Tutorial

What happens when the sizes do not match

Adding a 1×2 matrix to a 1×3 matrix is not allowed.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Try to add [A] and [B].
2ndMATRIX1+2ndMATRIX2ENTER

Addition, Subtraction, Multiplicationp. 151

Operation	Syntax	Rule
Add	`matrixA+matrixB`	Both matrices must be the same size. The answer holds the sums of corresponding elements.
Subtract	`matrixA−matrixB`	Both matrices must be the same size. The answer holds the differences of corresponding elements.
Multiply two matrices	`matrixA×matrixB`	The number of columns of matrixA must equal the number of rows of matrixB.
Multiply by a number	`matrix×value` or `value×matrix`	Every element is multiplied by the value.

Tutorial

Add, subtract and multiply two matrices

Matrix [A] is [[2,3][5,8]] and [B] is [[4,3][2,1]].

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Add the matrices: [A] + [B].
2ndMATRIX1+2ndMATRIX2ENTER
Subtract: [A] − [B].
2ndMATRIX1−2ndMATRIX2ENTER
Multiply: [A] × [B]. Row-by-column products are added up for each cell.
2ndMATRIX1×2ndMATRIX2ENTER
Multiply by a number: 3 × [A].
3×2ndMATRIX1ENTER
The order of a number and a matrix does not matter: [A] × 3 gives the same result.
2ndMATRIX1×3ENTER

Negationp. 152

Putting the negation key (−) in front of a matrix, ⁻matrix, changes the sign of every element.

Tutorial

Negate a matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press (−), then paste [A] and calculate.
(−)2ndMATRIX1ENTER

abs(p. 152

abs(

abs(matrix)

Absolute value (MATH NUM menu). With a matrix it returns a new matrix holding the absolute value of every element.

Keys: MATH▶︎1

abs([[⁻1,2][3,⁻4]]) returns [[1,2][3,4]]

Tutorial

Absolute value of every element

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the MATH menu, move to the NUM tab and choose 1:abs(.
MATH▶︎1
Paste [A], close the parenthesis and calculate.
2ndMATRIX1)ENTER

round(p. 152

round(

round(matrix[,#decimals])

Rounds every element of a matrix to #decimals places (9 or fewer). If you leave #decimals out, elements are rounded to 10 digits. Returns a new matrix.

Keys: MATH▶︎2

round([[1.234,5.678][9.012,3.456]],1) returns [[1.2,5.7][9,3.5]]

Tutorial

Round every element to one decimal place

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 2:round( from the MATH NUM menu.
MATH▶︎2
Paste [A], type a comma and the number of decimals (1), close the parenthesis and calculate.
2ndMATRIX1,1)ENTER

Inversep. 152

matrix⁻¹

matrix⁻¹

Inverts a square matrix. Use the x⁻¹ key (x⁻¹), or ^ followed by ⁻1. The matrix must be square and its determinant must not be zero, otherwise the calculator reports an error.

Keys: 2ndMATRIX1x⁻¹

[[2,3][5,8]]⁻¹ returns [[8,⁻3][⁻5,2]]

Tutorial

Invert a matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Paste [A] and press the x⁻¹ key (x⁻¹), then calculate.
2ndMATRIX1x⁻¹ENTER
Check it: multiplying [A] by the inverse you just found (it is in Ans, pasted with 2ndANS) gives the identity matrix.
2ndMATRIX1×2ndANSENTER

Note A matrix whose determinant is zero (a singular matrix) cannot be inverted. The calculator shows ERR:SINGULAR MAT.

Tutorial

Inverting a singular matrix gives an error

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Try to invert [A]. Its second row is twice its first, so the determinant is 0.
2ndMATRIX1x⁻¹ENTER

Powersp. 153

A matrix can be raised to an integer power from 0 to 255, provided it is square. Use x² for the square, 3:³ in the MATH menu for the cube, or ^ followed by the power for any other exponent.

Form	Keys
`matrix²`	`x²`
`matrix³`	`MATH3`
`matrix^power`	`^`

The 0th power of a square matrix is the identity matrix.

Tutorial

Square, cube and raise a matrix to a power

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Square [A] with the x² key.
2ndMATRIX1x²ENTER
Cube it with 3:³ from the MATH menu.
2ndMATRIX1MATH3ENTER
Use the ^ key for any other power, for example [A]^4.
2ndMATRIX1^4ENTER

Relational Operationsp. 153

You can compare two matrices with = and ≠ from the TEST menu (2ndTEST). They must be the same size, and the comparison is made element by element. The other relational operators (<, >, ≤, ≥) are not valid with matrices.

Expression	Result
`matrixA=matrixB`	1 if every pair of elements is equal; 0 if any pair differs
`matrixA≠matrixB`	1 if at least one pair differs; 0 if all pairs are equal

Tutorial

Compare two matrices

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Type [A], open the TEST menu and choose 1:=, then paste [B] and calculate. One element differs, so the answer is 0.
2ndMATRIX12ndTEST12ndMATRIX2ENTER
Now ask whether they are not equal, with 2:≠ in the TEST menu. The answer is 1.
2ndMATRIX12ndTEST22ndMATRIX2ENTER

iPart(, fPart(, int(p. 154

These three functions live in the MATH NUM menu and work on every element of a matrix.

iPart(

iPart(matrix)

Integer part: returns a matrix holding the integer part of each element (the digits before the decimal point).

Keys: MATH▶︎3

iPart([[1.5,⁻2.5]]) returns [[1,⁻2]]

fPart(

fPart(matrix)

Fractional part: returns a matrix holding the fractional part of each element.

Keys: MATH▶︎4

fPart([[1.5,⁻2.5]]) returns [[.5,⁻.5]]

int(

int(matrix)

Greatest integer: returns a matrix holding the greatest integer that is less than or equal to each element.

Keys: MATH▶︎5

int([[1.5,⁻2.5]]) returns [[1,⁻3]]

Tutorial

Integer part, fractional part and greatest integer

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Take the integer part with 3:iPart( in the MATH NUM menu.
MATH▶︎32ndMATRIX1)ENTER
Take the fractional part with 4:fPart(.
MATH▶︎42ndMATRIX1)ENTER
Take the greatest integer with 5:int(. Note that int(⁻2.5) is ⁻3, not ⁻2.
MATH▶︎52ndMATRIX1)ENTER

Using the MATRX MATH Operationsp. 154

The MATH tab of the MATRX menu collects the functions that work only on matrices. Open it with 2ndMATRIX then ▶︎. The list is long enough to scroll. Item 10 is chosen with 0, and the items after it are chosen with the letter keys A to F.

1:det(	Calculates the determinant
2:ᵀ	Transposes the matrix
3:dim(	Returns the matrix dimensions
4:Fill(	Fills all elements with a constant
5:identity(	Returns the identity matrix
6:randM(	Returns a random matrix
7:augment(	Appends two matrices
8:Matr►list(	Stores a matrix to a list
9:List►matr(	Stores a list to a matrix
0:cumSum(	Returns the cumulative sums of a matrix
A:ref(	Returns the row-echelon form of a matrix
B:rref(	Returns the reduced row-echelon form
C:rowSwap(	Swaps two rows of a matrix
D:row+(	Adds two rows; stores in the second row
E:*row(	Multiplies a row by a number
F:*row+(	Multiplies a row, adds it to another row and stores in the second row

Tip In the MATH tab, press ▲︎ from the first item to wrap around to the end of the list. Items A to F are chosen with the letter keys, for example ALPHAB for B:rref(.

det(p. 155

det(

det(matrix)

Returns the determinant of a square matrix, which is a real number.

Keys: 2ndMATRIX▶︎1

det([[1,2][3,4]]) returns ⁻2

Tutorial

Find a determinant

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the MATRX MATH tab and choose 1:det(.
2ndMATRIX▶︎1
Paste [A], close the parenthesis and calculate.
2ndMATRIX1)ENTER

Transposep. 155

ᵀ

matrixᵀ

Transpose: returns a matrix in which each element (row, column) is swapped with the element (column, row). The rows of the original become the columns of the answer. Type it after the matrix.

Keys: 2ndMATRIX▶︎2

[[1,2,3][4,5,6]]ᵀ returns [[1,4][2,5][3,6]]

Tutorial

Transpose a matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Paste [A] first, because ᵀ comes after the matrix.
2ndMATRIX1
Open the MATRX MATH tab and choose 2:ᵀ, then calculate.
2ndMATRIX▶︎2ENTER

Accessing Matrix Dimensions with dim(p. 155

dim(

dim(matrix){rows,columns}→dim(matrixname)

With a matrix it returns a list with the dimensions, {rows columns}. Used on the right of STO► it creates or resizes a matrix (see the next two sections).

Keys: 2ndMATRIX▶︎3

dim([[1,2,3][4,5,6]]) returns {2 3}

To get just the number of rows or just the number of columns, store the dimensions in a list and read one entry. For example, dim([A])→L₁ and then L₁(1) is the number of rows and L₁(2) is the number of columns.

Tutorial

Find the size of a matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 3:dim( from the MATRX MATH tab.
2ndMATRIX▶︎3
Paste [A], close the parenthesis and calculate. The answer is the list {2 3}: 2 rows and 3 columns.
2ndMATRIX1)ENTER

Tutorial

Read the number of rows and columns separately

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Store the dimension list in L₁: type dim([A]), press STO► and 2ndL1 (L₁).
2ndMATRIX▶︎32ndMATRIX1)STO►2ndL1ENTER
L₁(1) is the number of rows.
2ndL1(1)ENTER
L₁(2) is the number of columns.
2ndL1(2)ENTER

Creating a Matrix with dim(p. 156

Store a two-item list {rows,columns} into dim( of a matrix name to create a new matrix of that size, with 0 in every cell.

{rows,columns}→dim(matrixname)

Tutorial

Create a 2×3 matrix [D] full of zeros

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Type the list {2,3} and press STO►.
2nd{2,32nd}STO►
Choose 3:dim( from the MATRX MATH tab, paste [D] from the NAMES menu, close the parenthesis and press ENTER.
2ndMATRIX▶︎32ndMATRIX4)ENTER

Redimensioning a Matrix with dim(p. 156

The same command applied to an existing matrix resizes it. Elements that fit inside the new size are kept, new cells are filled with 0, and elements outside the new size are deleted.

{rows,columns}→dim(matrixname)

Tutorial

Change [A] from 2×3 to 3×2

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Type the new size as a list, {3,2}, and press STO►.
2nd{3,22nd}STO►
Choose 3:dim(, paste [A], close the parenthesis and press ENTER.
2ndMATRIX▶︎32ndMATRIX1)ENTER
The elements that still fit (1, 2 and 4) were kept. The 3 and 6 that were in column 3 are gone, and the new third row is all zeros.

Fill(p. 156

Fill(

Fill(value,matrixname)

Stores the value in every element of the named matrix, replacing what was there.

Keys: 2ndMATRIX▶︎4

Fill(7,[A]) makes every element of [A] equal to 7

Tutorial

Fill a matrix with a constant

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 4:Fill( from the MATRX MATH tab.
2ndMATRIX▶︎4
Type the value 7, a comma, paste [A], close the parenthesis and press ENTER.
7,2ndMATRIX1)ENTER

identity(p. 156

identity(

identity(dimension)

Returns the identity matrix, a square matrix with 1 on the main diagonal and 0 everywhere else. The dimension is both the number of rows and the number of columns.

Keys: 2ndMATRIX▶︎5

identity(3) returns [[1,0,0][0,1,0][0,0,1]]

Tutorial

Create a 3×3 identity matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 5:identity( from the MATRX MATH tab.
2ndMATRIX▶︎5
Type 3, close the parenthesis and calculate.
3)ENTER

randM(p. 157

randM(

randM(rows,columns)

Creates a random matrix with the given number of rows and columns. Each element is a random integer from ⁻9 to 9. The seed stored to the rand function controls the values (see Chapter 2).

Keys: 2ndMATRIX▶︎6

randM(2,3) returns a 2×3 matrix such as [[8,8,⁻7][0,⁻2,4]]

Tutorial

Make a random 2×3 matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 6:randM( from the MATRX MATH tab.
2ndMATRIX▶︎6
Type 2, a comma, 3, close the parenthesis and calculate. Your numbers will differ each time.
2,3)ENTER

augment(p. 157

augment(

augment(matrixA,matrixB)

Appends matrixB to matrixA as new columns. The two matrices must have the same number of rows.

Keys: 2ndMATRIX▶︎7

augment([[1,2][3,4]],[[5][6]]) returns [[1,2,5][3,4,6]]

Tutorial

Join two matrices side by side

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 7:augment( from the MATRX MATH tab.
2ndMATRIX▶︎7
Paste [A], a comma, [B], close the parenthesis and calculate. The result has 4 columns.
2ndMATRIX1,2ndMATRIX2)ENTER

Matr►list(p. 157

Matr►list(

Matr►list(matrix,listnameA,...,listname n)Matr►list(matrix,column#,listname)

The first form fills each named list with one column of the matrix, column 1 into the first list, column 2 into the second, and so on. Extra list names, and extra matrix columns, are ignored. The second form copies one chosen column (column#) of the matrix into a single list.

Keys: 2ndMATRIX▶︎8

Matr►list([A],L₁,L₂)
Matr►list([A],2,L₃) fills L₃ with column 2 of [A]

Tutorial

Copy the columns of a matrix into lists

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 8:Matr►list( from the MATRX MATH tab.
2ndMATRIX▶︎8
Paste [A], then name the lists L₁ and L₂ (2ndL1 and 2ndL2), separated by commas, close the parenthesis and press ENTER.
2ndMATRIX1,2ndL1,2ndL2)ENTER
To copy one particular column, type the column number after the matrix. This puts column 2 of [A] into L₃.
2ndMATRIX▶︎82ndMATRIX1,2,2ndL3)ENTER

List►matr(p. 158

List►matr(

List►matr(listA,...,list n,matrixname)

Fills the named matrix column by column with the elements of each list. If the lists are not all the same length, each missing entry is filled with 0. Complex lists are not allowed.

Keys: 2ndMATRIX▶︎9

List►matr(L₁,L₂,[C]) makes the lists the first and second columns of [C]

Tutorial

Build a matrix from two lists

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 9:List►matr( from the MATRX MATH tab.
2ndMATRIX▶︎9
Type the list names L₁ and L₂, separated by a comma, then another comma and the destination, [C] from the NAMES menu. Close the parenthesis and press ENTER.
2ndL1,2ndL2,2ndMATRIX3)ENTER

cumSum(p. 158

cumSum(

cumSum(matrix)

Returns the cumulative sums of the elements, starting with the first element. Each column is totalled separately, from top to bottom.

Keys: 2ndMATRIX▶︎0

cumSum([[1,2][3,4]]) returns [[1,2][4,6]]

Tutorial

Cumulative sums down each column

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose 0:cumSum( from the MATRX MATH tab.
2ndMATRIX▶︎0
Paste [A], close the parenthesis and calculate. Column 1 becomes 1, 1+3 = 4 and column 2 becomes 2, 2+4 = 6.
2ndMATRIX1)ENTER

Row Operationsp. 158

Items A to F of the MATRX MATH menu are row operations. You can use them inside expressions, and you can enter any row number or value as an expression. Select the matrix from the MATRX NAMES menu.

Note A row operation returns a new matrix and does not change the matrix stored in memory. To keep the result, store it with STO►.

ref(, rref(p. 159

ref(

ref(matrix)

Returns the row-echelon form of a real matrix. The number of columns must be greater than or equal to the number of rows.

Keys: 2ndMATRIX▶︎ALPHAA

ref([[1,2,3,3][2,3,4,3]]) returns [[1,1.5,2,1.5][0,1,2,3]]

rref(

rref(matrix)

Returns the reduced row-echelon form of a real matrix. The number of columns must be greater than or equal to the number of rows. It is the quickest way to solve a system of linear equations (see the systems example).

Keys: 2ndMATRIX▶︎ALPHAB

rref([[1,2,3,3][2,3,4,3]]) returns [[1,0,⁻1,⁻3][0,1,2,3]]

Tutorial

Row-echelon and reduced row-echelon form

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose A:ref( from the MATRX MATH tab.
2ndMATRIX▶︎ALPHAA
Paste [A], close the parenthesis and calculate. Each row starts with a leading 1 further right than the row above.
2ndMATRIX1)ENTER
Now choose B:rref( and do the same. The leading 1s are the only non-zero entries in their columns.
2ndMATRIX▶︎ALPHAB2ndMATRIX1)ENTER

rowSwap(p. 159

rowSwap(

rowSwap(matrix,rowA,rowB)

Returns a matrix with rowA and rowB of the given matrix exchanged.

Keys: 2ndMATRIX▶︎ALPHAC

rowSwap([A],1,2) swaps the first and second rows of [A]

Tutorial

Swap two rows

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose C:rowSwap( from the MATRX MATH tab.
2ndMATRIX▶︎ALPHAC
Paste [A], then type the two row numbers 1 and 2, separated by commas, and calculate.
2ndMATRIX1,1,2)ENTER

row+(p. 159

row+(

row+(matrix,rowA,rowB)

Row addition: adds rowA and rowB of the matrix and puts the sum in rowB. The answer is a new matrix, and rowA is unchanged.

Keys: 2ndMATRIX▶︎ALPHAD

row+([A],1,2) adds row 1 to row 2

Tutorial

Add one row to another

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose D:row+( from the MATRX MATH tab.
2ndMATRIX▶︎ALPHAD
Paste [A], then rows 1 and 2, and calculate. Row 2 becomes 5+2 and 8+3.
2ndMATRIX1,1,2)ENTER

*row(p. 160

*row(

*row(value,matrix,row)

Row multiplication: multiplies the given row of the matrix by the value and puts the result back in that row. The answer is a new matrix.

Keys: 2ndMATRIX▶︎ALPHAE

*row(2,[A],1) doubles the first row of [A]

Tutorial

Multiply a row by a number

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose *E:row(** from the MATRX MATH tab.
2ndMATRIX▶︎ALPHAE
Type the multiplier 2, then [A], then the row number 1, and calculate.
2,2ndMATRIX1,1)ENTER

*row+(p. 160

*row+(

*row+(value,matrix,rowA,rowB)

Row multiplication and addition: multiplies rowA of the matrix by the value, adds the product to rowB, and stores the result in rowB. The answer is a new matrix.

Keys: 2ndMATRIX▶︎ALPHAF

*row+(⁻2,[A],1,2) adds ⁻2 times row 1 to row 2

Tutorial

Multiply a row and add it to another

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose *F:row+(** from the MATRX MATH tab.
2ndMATRIX▶︎ALPHAF
Type the multiplier 2, then [A], then row 1 and row 2, and calculate. Row 2 becomes 2×(2, 3) + (5, 8) = (9, 14).
2,2ndMATRIX1,1,2)ENTER

1:[A]	Edit matrix [A] (its current size is shown beside the name)
2:[B]	Edit matrix [B]
3:[C]	Edit matrix [C]
4:[D]	Edit matrix [D]
5:[E]	Edit matrix [E]
6:[F]	Edit matrix [F]
7:[G]	Edit matrix [G]
8:[H]	Edit matrix [H]
9:[I]	Edit matrix [I]
0:[J]	Edit matrix [J]

1:[A]	Paste the name [A] into the expression you are typing
2:[B]	Paste [B]
3:[C]	Paste [C]
4:[D]	Paste [D]
5:[E]	Paste [E]
6:[F]	Paste [F]
7:[G]	Paste [G]
8:[H]	Paste [H]
9:[I]	Paste [I]
0:[J]	Paste [J]