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

Given the matrices $A = \begin{pmatrix} x & 3 \\ 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , what is the value of $x$ such that the determinant of the product matrix $AB$ is $25$ ?

4Answer
B
2
C
-1
D
11.5

Answer

The value of

x

is 4.

The determinant of a product of matrices equals the product of their determinants,

\det(AB) = \det(A)\det(B)

. Since

\det(B) = 5

and

\det(A) = 2x - 3

, setting

5(2x - 3) = 25

yields

2x - 3 = 5

, giving

x = 4

Step-by-Step Solution

Calculate the determinant of matrix B

\det(B) = (2)(4) - (1)(3) = 8 - 3 = 5

The determinant of a 2x2 matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is given by

ad - bc

Express the determinant of matrix A in terms of x

\det(A) = (x)(2) - (3)(1) = 2x - 3

Apply the 2x2 determinant formula to matrix A.

Use the determinant product rule det(AB) = det(A) * det(B)

\det(AB) = (2x - 3) \cdot 5 = 10x - 15

The determinant of the product of two square matrices equals the product of their individual determinants.

Set det(AB) equal to 25 and solve for x

10x - 15 = 25 \implies 10x = 40 \implies x = 4

Equate the determinant expression to the given value 25 to isolate x.

Key Concept

Determinant of a Matrix Product

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