Given the matrices A = \begin{pmatrix} 3 & y \\ 2 & 1 \end{pmatrix} and B = \begin{pmatrix… — JAMB UTME

Given the matrices $A = \begin{pmatrix} 3 & y \\ 2 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ x & 2 \end{pmatrix}$ , if $AB = \begin{pmatrix} 9 & 16 \\ 5 & 10 \end{pmatrix}$ , what is the value of $x + y$ ?

$5$ Answer
B
$6$
C
$7$
D
$9$

Answer

The value of

x + y

5

Multiplying matrix

A

by matrix

B

using standard row-by-column multiplication yields

AB = \begin{pmatrix} 3 + xy & 12 + 2y \\ 2 + x & 10 \end{pmatrix}

. Setting this equal to

\begin{pmatrix} 9 & 16 \\ 5 & 10 \end{pmatrix}

gives

2 + x = 5 \implies x = 3

and

12 + 2y = 16 \implies y = 2

. Thus,

x + y = 3 + 2 = 5

Step-by-Step Solution

Compute the product matrix

AB

using matrix multiplication rules.

AB = \begin{pmatrix} 3(1) + y(x) & 3(4) + y(2) \\ 2(1) + 1(x) & 2(4) + 1(2) \end{pmatrix} = \begin{pmatrix} 3 + xy & 12 + 2y \\ 2 + x & 10 \end{pmatrix}

Matrix multiplication requires taking the dot product of rows from the first matrix and columns from the second matrix.

Equate the elements of

AB

with the given matrix

\begin{pmatrix} 9 & 16 \\ 5 & 10 \end{pmatrix}

to solve for

x

and

y

From row 2, column 1:

2 + x = 5 \implies x = 3

. From row 1, column 2:

12 + 2y = 16 \implies 2y = 4 \implies y = 2

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

Calculate the sum

x + y

x + y = 3 + 2 = 5

Substituting the values found for

x

and

y

into the requested expression.

Key Concept

Matrix Multiplication and Equality of Matrices

Alternative Method

Verification can be done by checking row 1, column 1:

3 + xy = 3 + (3)(2) = 9

, which matches the given matrix entry.

Estimated Time:1m 15s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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