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

Given the matrices $A = \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}$ , what is the smaller positive integer value of $k$ for which $\det(A^2 + B) = 4$ ?

5Answer
B
6
C
4
D
8

Answer

The smaller positive integer value of

k

is 5.

Evaluating

A^2

via matrix multiplication gives

\begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix}

. Adding

B

gives

A^2 + B = \begin{pmatrix} k^2 + 7 & 2k + 1 \\ 3k + 5 & 7 \end{pmatrix}

. Computing

\det(A^2 + B) = (k^2 + 7)(7) - (2k + 1)(3k + 5) = k^2 - 13k + 44

. Setting this to 4 results in

k^2 - 13k + 40 = 0

, which yields roots

k = 5

and

k = 8

. The smaller value is 5.

Step-by-Step Solution

Compute matrix

A^2

using matrix multiplication (row by column).

A^2 = \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix}

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

Add matrix

B

A^2

A^2 + B = \begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix} + \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} k^2 + 7 & 2k + 1 \\ 3k + 5 & 7 \end{pmatrix}

Matrix addition is performed by adding corresponding entries.

Calculate the determinant of

A^2 + B

and set it equal to 4.

\det(A^2 + B) = (k^2 + 7)(7) - (2k + 1)(3k + 5) = 7k^2 + 49 - (6k^2 + 13k + 5) = k^2 - 13k + 44 = 4

The determinant of a

2 \times 2

matrix

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

ad - bc

Solve the quadratic equation for

k

k^2 - 13k + 40 = 0 \implies (k - 5)(k - 8) = 0 \implies k = 5 \text{ or } k = 8

Factoring the quadratic equation gives the two possible integer roots.

Key Concept

Matrix multiplication, matrix addition, determinant of a 2x2 matrix, and quadratic equation solving.

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