Given that the matrix A = \begin{pmatrix} x & 2 \\ 5 & x - 3 \end{pmatrix} is a singular m… — JAMB UTME

Given that the matrix $A = \begin{pmatrix} x & 2 \\ 5 & x - 3 \end{pmatrix}$ is a singular matrix, what is the positive value of $x$ ?

The positive value of

x

5

For a matrix to be singular, its determinant must be zero. Calculating the determinant of

A

yields

\det(A) = x(x - 3) - (2)(5) = x^2 - 3x - 10

. Setting this to zero and factoring gives

(x - 5)(x + 2) = 0

, leading to

x = 5

x = -2

. The positive solution is

5

Set the determinant of the matrix equal to zero.

\det(A) = (x)(x - 3) - (2)(5) = 0

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

Expand and simplify the algebraic equation.

x^2 - 3x - 10 = 0

Expanding

x(x - 3)

gives

x^2 - 3x

, and subtracting

10

forms a standard quadratic equation.

Factor the quadratic equation to solve for

x

(x - 5)(x + 2) = 0 \implies x = 5 \text{ or } x = -2

The factors of

-10

that sum to

-3

are

-5

and

+2

Select the positive value requested by the question.

x = 5

The question specifically asks for the positive value of

x

Singular Matrices and Determinants

Question