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

Given the matrices $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & k \end{pmatrix}$ , find the value of $k$ such that the determinant of the product matrix $AB$ is equal to 50.

Answer: 3

Answer

The value of

k

is 3.

By applying the determinant product rule

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

, we find

\det(A) = (4)(3) - (2)(1) = 10

. Given

\det(AB) = 50

, it follows that

\det(B) = 50 / 10 = 5

. Since

\det(B) = (1)(k) - (-1)(2) = k + 2

, setting

k + 2 = 5

gives

k = 3

Step-by-Step Solution

Calculate the determinant of matrix A

\det(A) = (4 \times 3) - (2 \times 1) = 12 - 2 = 10

The determinant of a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} is ad - bc.

Apply the product property of determinants

\det(B) = \frac{\det(AB)}{\det(A)} = \frac{50}{10} = 5

For any two square matrices of the same dimension, \det(AB) = \det(A) \cdot \det(B).

Express the determinant of matrix B in terms of k and solve

\det(B) = (1)(k) - (-1)(2) = k + 2 = 5 \implies k = 3

Equating the calculated determinant formula for B to its numerical value of 5.

Key Concept

Determinant Product Property and 2x2 Matrix Determinant

Estimated Time:1m 30s

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