Matrices A, B, and C are defined such that A = \begin{bmatrix} x & 3 \\ -2 & 4 \end{bmatri… — ACT

Matrices $A$ , $B$ , and $C$ are defined such that $A = \begin{bmatrix} x & 3 \\ -2 & 4 \end{bmatrix}$ , $B = \begin{bmatrix} 2 & -1 \\ y & 5 \end{bmatrix}$ , and $C = \begin{bmatrix} 14 & 11 \\ 4 & 22 \end{bmatrix}$ . If $AB = C$ , what is the value of $x + y$ ?

Answer: 6

Answer

The value of

x + y

is 6.

The correct answer is 6 because performing the matrix multiplication

AB

yields the matrix

\begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

. Equating this to matrix

C

gives the equations

-x + 15 = 11

, which yields

x = 4

, and

-4 + 4y = 4

, which yields

y = 2

. The sum of these values is

4 + 2 = 6

. This is verified by checking the remaining equation

2(4) + 3(2) = 14

Step-by-Step Solution

Multiply matrix

A

and matrix

B

to express the product

AB

algebraically.

AB = \begin{bmatrix} x(2) + 3(y) & x(-1) + 3(5) \\ -2(2) + 4(y) & -2(-1) + 4(5) \end{bmatrix} = \begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

To solve for the variables, we must first determine the resulting matrix from the multiplication of

A

and

B

Set up equations by equating the entries of

AB

with the corresponding entries of matrix

C

2x + 3y = 14

-x + 15 = 11

, and

-4 + 4y = 4

Two matrices are equal if and only if all of their corresponding elements are equal.

Solve the single-variable equation from the first row, second column for

x

-x + 15 = 11 \implies -x = -4 \implies x = 4

Isolating

x

gives its numeric value.

Solve the single-variable equation from the second row, first column for

y

-4 + 4y = 4 \implies 4y = 8 \implies y = 2

Isolating

y

gives its numeric value.

Verify consistency with the remaining equation and calculate the final sum

x + y

2(4) + 3(2) = 8 + 6 = 14

(consistent), and

x + y = 4 + 2 = 6

Checking the first row, first column confirms the values are correct, and adding them yields the requested sum.

Key Concept

Matrix multiplication and matrix equality

Question

Answer

Step-by-Step Solution

Key Concept

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