Understanding Singular Matrices and Determinants

The question asks us to find the specific value of x that will make the given matrix singular. A matrix is defined as singular if its determinant is equal to zero.

Matrix Representation

The matrix provided is:

Let's denote this matrix as A.

Calculating the Determinant

To find the value of x that makes the matrix singular, we first need to calculate its determinant, denoted as $ \det(A) $. We can use the cofactor expansion method. Let's expand along the first row:

Now, we calculate the determinants of the 2x2 matrices:

• $$ \begin{vmatrix} 0 & 2 \\ 6 & 0 \end{vmatrix} = (0 \times 0) - (2 \times 6) = 0 - 12 = -12 $$
• $$ \begin{vmatrix} 4 & 2 \\ 12 & 0 \end{vmatrix} = (4 \times 0) - (2 \times 12) = 0 - 24 = -24 $$
• $$ \begin{vmatrix} 4 & 0 \\ 12 & 6 \end{vmatrix} = (4 \times 6) - (0 \times 12) = 24 - 0 = 24 $$

Substitute these values back into the determinant formula:

Solving for x to Achieve Singularity

For the matrix to be singular, its determinant must be zero. So, we set $ \det(A) = 0 $:

Now, we solve this linear equation for x:

1. Add 96 to both sides: $$ 24x = 96 $$
2. Divide both sides by 24: $$ x = \frac{96}{24} $$
3. Calculate the value: $$ x = 4 $$

Therefore, the value of x that makes the matrix singular is 4.