Given the matrices A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} and B = \begin{pmatri… — JAMB UTME

Given the matrices $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & x \\ -2 & 5 \end{pmatrix}$ , if the determinant of the product matrix $AB$ is equal to $121$ , find the value of $x$ .

Answer: 3

Answer

The value of

x

3

By applying the property

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

, we compute

\det(A) = (2)(4) - (-1)(3) = 11

and

\det(B) = (1)(5) - (x)(-2) = 5 + 2x

. Substituting into the equation gives

11(5 + 2x) = 121

, which yields

5 + 2x = 11

and leads to

x = 3

Step-by-Step Solution

Find the determinant of matrix

A

\det(A) = (2)(4) - (-1)(3) = 11

The determinant of a

2 \times 2

matrix

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

is calculated as

ad - bc

Express the determinant of matrix

B

in terms of

x

\det(B) = (1)(5) - (x)(-2) = 5 + 2x

Apply the

2 \times 2

determinant formula to matrix

B

Apply the determinant product rule

\det(AB) = \det(A) \cdot \det(B) = 11(5 + 2x) = 121

For any square matrices

A

and

B

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

Solve the resulting linear equation for

x

x = 3

Dividing

121

11

gives

5 + 2x = 11

, which simplifies to

2x = 6

and

x = 3

Key Concept

Determinant of a Matrix Product

Estimated Time:1m 30s

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