Question

1. what is the missing value of $x$ that makes the matrix equation true?$\begin{bmatrix}1 & 0 & -6 \\4 & -1 & x \\-5 & 0 & 3end{bmatrix} cdot \begin{bmatrix}1 & -3 & 2 \\0 & -4 & 1 \\-2 & 5 & 0end{bmatrix} = \begin{bmatrix}13 & -33 & 2 \\0 & 2 & 7 \\-11 & 30 & -10end{bmatrix}$answer:9) given $a = \begin{bmatrix}-2 & 4 & 0 \\-3 & 5 & 1 \\-4 & -4 & 2end{bmatrix}$ and $b = \begin{bmatrix}-5 & 1 & 1 \\4 & -2 & 2 \\3 & -1 & 3end{bmatrix}$when $ab$ is multiplied, what element (number) is in $e_{22}$?answer:10) given $c = \begin{bmatrix}1 & -3 \\5 & -4 \\2 & 1end{bmatrix}$ and $d = \begin{bmatrix}-4 & 0 & 3 \\2 & -3 & 4end{bmatrix}$when $cd$ is multiplied, what element (number) is in $e_{13}$?answer:

Explanation:

(Problem 8):

Step1: Pick matching matrix entry

Use the entry that involves $x$: the 2nd row, 3rd column of the product matrix, which is 7.

Step2: Compute row-column dot product

The 2nd row of first matrix

$$\begin{bmatrix}4 & -1 & x\end{bmatrix}$$

dotted with 3rd column of second matrix

$$\begin{bmatrix}2 \\ 1 \\ 0\end{bmatrix}$$

:
$4(2) + (-1)(1) + x(0) = 7$

Step3: Simplify and solve for $x$

$8 - 1 + 0 = 7$
This simplifies to $7=7$, so use another entry with $x$: 2nd row, 2nd column of product (value 2):
$4(-3) + (-1)(-4) + x(5) = 2$
$ -12 + 4 + 5x = 2$
$ -8 + 5x = 2$
$5x = 10$
$x = \frac{10}{5} = 2$

(Problem 9):

Step1: Identify target entry $e_{22}$

This is the entry from 2nd row of $A$ and 2nd column of $B$.

Step2: Compute row-column dot product

2nd row of $A$:

$$\begin{bmatrix}-3 & 5 & 1\end{bmatrix}$$

, 2nd column of $B$:

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

$(-3)(1) + 5(-2) + 1(-1)$

Step3: Calculate the result

$-3 -10 -1 = -14$

(Problem 10):

Step1: Identify target entry $e_{13}$

This is the entry from 1st row of $C$ and 3rd column of $D$.

Step2: Compute row-column dot product

1st row of $C$:

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

, 3rd column of $D$:

$$\begin{bmatrix}3 \\ 4\end{bmatrix}$$

$1(3) + (-3)(4)$

Step3: Calculate the result

$3 - 12 = -9$

Answer:

1. Problem 8: $2$
2. Problem 9: $-14$
3. Problem 10: $-9$