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

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

$-3$ Answer
B
$10$
C
$-1$
D
$1$

Answer

The value of

x + y

-3

Evaluating the matrix products

AB

and

BA

using standard matrix multiplication rules yields

AB = \begin{pmatrix} 2+2x & 8+xy \\ 5 & -4+3y \end{pmatrix}

and

BA = \begin{pmatrix} -2 & x+12 \\ 4-y & 2x+3y \end{pmatrix}

. Equating entry-by-entry gives

2+2x = -2 \implies x = -2

and

5 = 4-y \implies y = -1

. Adding these together gives

x + y = -3

Step-by-Step Solution

Compute the product matrix

AB

AB = \begin{pmatrix} 2(1) + x(2) & 2(4) + x(y) \\ -1(1) + 3(2) & -1(4) + 3(y) \end{pmatrix} = \begin{pmatrix} 2 + 2x & 8 + xy \\ 5 & -4 + 3y \end{pmatrix}

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

Compute the product matrix

BA

BA = \begin{pmatrix} 1(2) + 4(-1) & 1(x) + 4(3) \\ 2(2) + y(-1) & 2(x) + y(3) \end{pmatrix} = \begin{pmatrix} -2 & x + 12 \\ 4 - y & 2x + 3y \end{pmatrix}

Evaluate

BA

by multiplying rows of

B

by columns of

A

Equate corresponding entries of

AB

and

BA

since

AB = BA

From row 1, col 1:

2 + 2x = -2 \implies 2x = -4 \implies x = -2

.
From row 2, col 1:

5 = 4 - y \implies y = -1

Two matrices are equal if and only if all corresponding entries are equal.

Verify consistency on remaining entries and calculate

x + y

Row 1, col 2 check:

8 + (-2)(-1) = 10

and

-2 + 12 = 10

.
Row 2, col 2 check:

-4 + 3(-1) = -7

and

2(-2) + 3(-1) = -7

.
Sum:

x + y = -2 + (-1) = -3

Verifying consistency ensures the system of equations has a valid unique solution.

Key Concept

Matrix Multiplication Commutativity and Matrix Equality

Estimated Time:2m 30s

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