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. $1$
• D. 32

Answer 1

The Correct Option is C

To find the value of \(x\) for which matrix \( A \) is singular, we must compute the determinant of matrix \( A \) and set it to zero. A singular matrix is one with a determinant of zero. Given the matrix \( A = \begin{bmatrix} 2 & 4 \\ x & 2 \end{bmatrix} \), the formula to calculate the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is \(ad - bc\).

For matrix \(A\), we have:

• \(a = 2\)
• \(b = 4\)
• \(c = x\)
• \(d = 2\)

Calculate the determinant of \( A \):

\[\text{det}(A) = (2)(2) - (4)(x) = 4 - 4x\]

Set the determinant to zero for matrix \(A\) to be singular:

\[4 - 4x = 0\]

Solving for \(x\):

\[4 = 4x\]

\[x = \frac{4}{4}\]

\[x = 1\]

Therefore, the value of \(x\) for which matrix \(A\) is singular is \(1\).

Answer 2

To determine the value of \( x \) that makes the matrix \( A \) singular, we need to calculate the determinant of \( A \) and set it equal to zero. A \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is singular if its determinant is zero. The determinant is calculated as follows: 

\(\text{det}(A) = ad - bc\)

For the given matrix \( A = \begin{bmatrix} 2 & 4 \\ x & 2 \end{bmatrix} \), the determinant is:

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

To find the value of \( x \) that makes \( A \) singular, set the determinant to zero:

\(4 - 4x = 0\)

Solving for \( x \), we get:

\(4 = 4x\)

\(x = \frac{4}{4} = 1\)

Therefore, the value of \( x \) that makes \( A \) singular is 1.