A 2 \times 2 matrix is given by A = \begin{pmatrix} x + 2 & 3 \\ 4 & x - 2 \end{pmatrix}. … — JAMB UTME

A $2 \times 2$ matrix is given by $A = \begin{pmatrix} x + 2 & 3 \\ 4 & x - 2 \end{pmatrix}$ . If $A$ is a singular matrix, what is the positive value of $x$ ?

Answer: 4

Answer

The positive value of

x

4

A matrix is singular when its determinant is zero. Expanding

\det(A) = (x+2)(x-2) - 12 = 0

yields

x^2 - 16 = 0

. Solving

x^2 = 16

gives

x = \pm 4

. The positive value is

4

Step-by-Step Solution

Apply the condition for a matrix to be singular

\det(A) = 0

By definition, a matrix is singular if and only if its determinant is zero.

Formulate the determinant equation for matrix

A

(x + 2)(x - 2) - (3)(4) = 0

The determinant of a

2 \times 2

matrix

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

is calculated as

ad - bc

Expand and simplify the algebraic equation

x^2 - 4 - 12 = 0 \implies x^2 - 16 = 0

Expanding

(x+2)(x-2)

gives

x^2 - 4

, and subtracting

12

yields

x^2 - 16

Solve for

x

and select the positive root

x^2 = 16 \implies x = 4 \text{ or } x = -4 \implies x = 4

Taking the square root gives two roots, and the problem explicitly specifies the positive value.

Key Concept

Singular Matrices and Determinants

Estimated Time:1m 15s

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