11.5 Matrices and Matrix Operations

Write each system of linear equations as an augmented matrix

ⓐ $\left\{\begin{matrix} 3 x - y = - 1 \\ 2 y = 2 x + 5 \end{matrix}\right$

ⓑ $\left\{\begin{matrix} 4 x + 3 y = - 2 \\ x - 2 y - 3 z = 7 \\ 2 x - y + 2 z = - 6 \end{matrix}\right$

ⓐ Add the two matrices $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}$

ⓑ Subtract the two matrices $A = \begin{bmatrix} 2 & - 4 \\ 5 & 3 \end{bmatrix} B = \begin{bmatrix} 6 & 9 \\ 7 & 8 \end{bmatrix}$

ⓒ Multiply the matrix $A = \begin{bmatrix} 2 & - 4 \\ 5 & 3 \end{bmatrix}$ by 5.

Given matrix $A :$

1. ⓐWhat are the dimensions of matrix $A ?$
2. ⓑWhat are the entries at $a_{31}$ and $a_{22} ?$

Find the sum of $A$ and $B ,$ given

Find the sum of $A$ and $B .$

Find the difference of $A$ and $B .$

Given $A$ and $B :$

1. ⓐFind the sum.
2. ⓑFind the difference.

Add matrix $A$ and matrix $B .$

Multiply matrix $A$ by the scalar 3.

Given matrix $B ,$ find $−2 B$ where

Find the sum $3 A + 2 B .$

Given $A$ and $B :$

1. ⓐ Find $A B .$
2. ⓑ Find $B A .$

Find $A B - C$ given

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

Can both the products $A B$ and $B A$ be defined? If so, explain how; if not, explain why.

Does matrix multiplication commute? That is, does $A B = B A ?$ If so, prove why it does. If not, explain why it does not.

$C + D$

$B - E$

$D - B$

$3 B$

$−4 C$

$100 D$

$B C$

$B D$

$C B$

$4 A + 5 D$

$3 D + 4 E$

$100 D −10 E$

$B A$

$B C$

$B^{2}$

$B^{2} A^{2}$

$\left(A B\right)^{2}$

$A B$

$B D$

$D^{2}$

$D^{3}$

$A \left(B C\right)$

$B A$

$B C$

$B^{2}$

$B^{4}$

Using the above questions, find a formula for $B^{n} .$ Test the formula for $B^{201}$ and $B^{202} ,$ using a calculator.