Matrices

18.1Create and store a matrix#

The calculator has ten matrices, [A] to [J]. On the home screen a matrix is written with square brackets: each row is in its own pair, so [[1,2][3,4]] has two rows. Use 2nd[× for [ and 2nd]− for ]. Store it with sto→ and a name from 2ndMATRIXx⁻¹ (MATRIX), NAMES tab.

Example 18.1

Store a 2×2 matrix in [A]

1. Type [[1,2][3,4]], press sto, choose [A] with 2nd inv, 1, and press enter.

   2nd[×2nd[×1,22nd]−2nd[×3,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter

The MATRIX menu has three tabs: NAMES (paste a matrix name), MATH (operations) and EDIT (fill in a matrix).

18.2Fill a matrix in the matrix editor#

Press 2ndMATRIXx⁻¹, ◀ for EDIT and choose a name. First type the number of rows and press enter, then the number of columns and enter. The cells appear: type each value and press enter to move along the row and then to the next row. Press clear to leave.

Example 18.2

Build [B] and take its determinant

$$\det\begin{bmatrix}4&7\\2&6\end{bmatrix}=24-14=10$$

1. Open EDIT, choose [B], and set 2 rows and 2 columns.

   2ndMATRIXx⁻¹◀22enter2enter

   

2. Enter 4, 7, 2, 6.

   4enter7enter2enter6enter

3. Go home and calculate det([B]).

   2ndQUITmode2ndMATRIXx⁻¹▶12ndMATRIXx⁻¹2)enter

18.3Add, multiply, invert and transpose#

Matrices of the same size can be added and subtracted. A matrix times a number scales every entry. Two matrices multiply when the columns of the first equal the rows of the second. x⁻¹ gives the inverse of a square matrix (a SINGULAR MAT error if there is none), and ᵀ in the MATRIX MATH menu (item 2) gives the transpose.

Example 18.3

Multiply [A] by itself

$$\begin{bmatrix}1&2\\3&4\end{bmatrix}^{2}=\begin{bmatrix}7&10\\15&22\end{bmatrix}$$

1. Store [[1,2][3,4]] in [A].

   2nd[×2nd[×1,22nd]−2nd[×3,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter

2. Type [A][A].

   2ndMATRIXx⁻¹12ndMATRIXx⁻¹1enter

Example 18.4

Inverse

$$A^{-1}=\begin{bmatrix}-2&1\\1.5&-0.5\end{bmatrix}$$

1. Store [[1,2][3,4]] in [A], then type [A] followed by the x⁻¹ key.

   2nd[×2nd[×1,22nd]−2nd[×3,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter2ndMATRIXx⁻¹1x⁻¹enter

Example 18.5

Transpose

1. Store [[1,2][3,4]] in [A]. Type [A], then 2nd inv, right, 2 for ᵀ.

   2nd[×2nd[×1,22nd]−2nd[×3,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter2ndMATRIXx⁻¹12ndMATRIXx⁻¹▶2enter

Example 18.6

Determinant

$$\det A=1\cdot4-2\cdot3=-2$$

1. Store the matrix, then use det( from the MATH tab (item 1).

   2nd[×2nd[×1,22nd]−2nd[×3,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter2ndMATRIXx⁻¹▶12ndMATRIXx⁻¹1)enter

18.4The MATRIX MATH menu#

Example 18.7

Identity matrix

1. Press 2nd inv, right, 5 (identity(), type 2 and a bracket.

   2ndMATRIXx⁻¹▶52)enter

Example 18.8

Size of a matrix

1. Store a matrix in [A]. Press 2nd inv, right, 3 (dim(), paste [A] and close.

   2nd[×2nd[×1,22nd]−2nd[×3,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter2ndMATRIXx⁻¹▶32ndMATRIXx⁻¹1)enter

18.5Solve a system of linear equations with rref#

Write the system as an augmented matrix, one row per equation, with the constants in the last column, and ask for rref(. The identity pattern on the left leaves the solutions in the last column.

Example 18.9

Solve 2x + y = 5 and x + 3y = 10

1. Press 2nd inv, right, alpha apps (B: rref(), then type the augmented matrix and a bracket.

   2ndMATRIXx⁻¹▶alphaBapps2nd[×2nd[×2,1,52nd]−2nd[×1,3,102nd]−2nd]−)enter

Example 18.10

Row-reduce a 2×3 matrix

1. rref( of [[1,2,3][4,5,6]].

   2ndMATRIXx⁻¹▶alphaBapps2nd[×2nd[×1,2,32nd]−2nd[×4,5,62nd]−2nd]−)enter

Example 18.11

No inverse

1. Store [[1,2][2,4]] in [A] (its rows are multiples) and try [A]⁻¹.

   2nd[×2nd[×1,22nd]−2nd[×2,42nd]−2nd]−sto→2ndMATRIXx⁻¹1enter2ndMATRIXx⁻¹1x⁻¹enter