For what value of x, matrix `[[6-x,4],[3-x,1]]` is a singular matrix ? A) 1 B) 2 C) -1 D) -2

Options

A. 1
B. 2
C. `-1`
D. `-2`

Answer

To determine the value of \( x \) for which the matrix

\[ A = \begin{bmatrix} 6-x & 4 \\ 3-x & 1 \end{bmatrix} \]

is a singular matrix, we need to find when the determinant of matrix \( A \) equals zero.

### Step 1: Write the determinant of the matrix

The determinant of a \( 2 \times 2 \) matrix

\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

is calculated using the formula:

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

For our matrix \( A \):

- \( a = 6 - x \)
- \( b = 4 \)
- \( c = 3 - x \)
- \( d = 1 \)

Thus, the determinant of matrix \( A \) is:

\[ \text{det}(A) = (6 - x) \cdot 1 - 4 \cdot (3 - x) \]

### Step 2: Set the determinant to zero

Since we want the matrix to be singular, we set the determinant equal to zero:

\[ (6 - x) - 4(3 - x) = 0 \]

### Step 3: Simplify the equation

Now, we simplify the equation:

\[ 6 - x - 12 + 4x = 0 \]

Combining like terms gives:

\[ 3x - 6 = 0 \]

### Step 4: Solve for \( x \)

Now, we solve for \( x \):

\[ 3x = 6 \]

Dividing both sides by 3:

\[ x = 2 \]

### Conclusion

Thus, the value of \( x \) for which the matrix is singular is

\[ \boxed{2} \]