Question

If $A = \begin{bmatrix} 2 & 4 \\\ x & 2 \end{bmatrix}$ and $A$ is singular, then $x$ is equal to:

• A. $\frac{1}{4}$
• B. -7
• C. $-\frac{1}{4}$
• D. 32

Answer 1

Answer: (C)

$-\frac{1}{4}$

Answer 2

To determine the value of $x$, we use the fact that $A$ is singular. A matrix is singular if its determinant is zero. The given matrix is:

$A = \begin{bmatrix} 2 & 4 \\\ x & -\frac{1}{2} \end{bmatrix}.$

The determinant of $A$ is:

$\text{det}(A) = (2)(-1) - (x)(4) = -2 - 4x.$

Set $\text{det}(A) = 0$ because $A$ is singular:

$-2 - 4x = 0.$

Solve for $x$:

$4x = -2 \implies x = -\frac{1}{4}.$

Thus, $x = -\frac{1}{4}$.