Matrices A and B are defined as: A = \begin{bmatrix} x & 4 \\ -3 & 2 \end{bmatrix}, \quad … — ACT

Matrices $A$ and $B$ are defined as:

A = \begin{bmatrix} x & 4 \\ -3 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}

Let $C$ represent the product matrix $AB$ , where:

C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}

If the element $c_{12}$ is equal to $2$ , what is the value of $x$ ?

Answer: 5

To find the entry

c_{12}

in the first row and second column of the product matrix

C = AB

, we calculate the dot product of the first row of matrix

A

and the second column of matrix

B

. The first row of

A

[x, 4]

and the second column of

B

[-2, 3]^T

. Thus,

c_{12} = x(-2) + 4(3) = -2x + 12

. Setting this equal to the given value of

2

gives the equation

-2x + 12 = 2

. Subtracting

12

from both sides results in

-2x = -10

. Dividing by

-2

yields

x = 5

Identify the formula for the element

c_{12}

in the product matrix

C = AB

c_{12} = -2x + 12

The element

c_{12}

is located in the first row and second column of the product matrix, so it is the product of the first row of

A

, which is

[x, 4]

, and the second column of

B

, which is

[-2, 3]^T

Set the expression for

c_{12}

equal to the given value of

2

and solve the linear equation for

x

x = 5

Setting

-2x + 12 = 2

leads to

-2x = -10

, and dividing both sides by

-2

yields

x = 5

Matrix Multiplication and Element-wise Operations

Question