Question

test – matrices
part 1: basic matrix operations. write \undefined\ for the expressions that are undefined.
given: $a = \

$$\begin{bmatrix}-7 & 9 \\\\ 10 & 2 \\\\ -1 & 3 \\end{bmatrix}$$

$ $b = \

$$\begin{bmatrix}-9 & -7 & 4 \\\\ 0 & 2 & -3 \\end{bmatrix}$$

$ $c = \

$$\begin{bmatrix}-1 & -1 \\\\ 8 & 2 \\\\ -4 & 6 \\end{bmatrix}$$

$ $d = \

$$\begin{bmatrix}-2 & 2 & -3 \\\\ -4 & 5 & 12 \\end{bmatrix}$$

$

1. $a + c$
2. $d - 3b$
3. $4a + b$
4. $\

$$\begin{bmatrix}2 & 3 \\\\ -1 & 0 \\end{bmatrix}$$

\times \

$$\begin{bmatrix}1 & -4 \\\\ -2 & 3 \\end{bmatrix}$$

$

1. $\

$$\begin{bmatrix}0 & 6 \\\\ -1 & 5 \\end{bmatrix}$$

\times \

$$\begin{bmatrix}10 & 7 \\\\ -8 & 1 \\end{bmatrix}$$

$

1. bonus $\

$$\begin{bmatrix}3 & 1 & 2 \\\\ 2 & -1 & -1 \\end{bmatrix}$$

\times \

$$\begin{bmatrix}-1 & 3 \\\\ -2 & 0 \\\\ 4 & -4 \\end{bmatrix}$$

$

Explanation:

1) \( A + C \)

Step1: Check matrix dimensions

Matrix \( A \) is \( 3 \times 2 \) (3 rows, 2 columns), matrix \( C \) is \( 3 \times 2 \). Since they have the same dimensions, addition is defined.

Step2: Add corresponding elements

$$
LATEXBLOCK0
$$

Step1: Find \( 3B \)

Matrix \( B \) is \( 2 \times 3 \), so \( 3B = 3

$$\begin{bmatrix}-9&-7&4\\0&2&-3\end{bmatrix}$$

=

$$\begin{bmatrix}-27&-21&12\\0&6&-9\end{bmatrix}$$

\)

Step2: Check dimensions for subtraction

Matrix \( D \) is \( 2 \times 3 \), \( 3B \) is \( 2 \times 3 \), so subtraction is defined.

Step3: Subtract corresponding elements

$$
LATEXBLOCK2
$$

Step1: Find \( 4A \)

Matrix \( A \) is \( 3 \times 2 \), \( 4A = 4

$$\begin{bmatrix}-7&9\\10&2\\-1&3\end{bmatrix}$$

=

$$\begin{bmatrix}-28&36\\40&8\\-4&12\end{bmatrix}$$

\)

Step2: Check dimensions for addition

Matrix \( 4A \) is \( 3 \times 2 \), matrix \( B \) is \( 2 \times 3 \). Different dimensions, so operation is undefined.

Answer:

\(

$$\begin{bmatrix}-8&8\\18&4\\-5&9\end{bmatrix}$$

\)

2) \( D - 3B \)