Question

If \( A = \begin{bmatrix} 2 - k & 2 \\ 1 & 3 - k \end{bmatrix} \) is a singular matrix, then the value of \( 5k - k^2 \) is:

• A. 0
• B. 6
• C. -6
• D. 4

Hint:

For a singular matrix, the determinant must be zero. Use this property to solve for unknowns in matrix problems.

Answer 1

The Correct Option is D

We are given the matrix \( A = \begin{bmatrix} 2 - k & 2 \\ 1 & 3 - k \end{bmatrix} \), and we need to find the value of \( 5k - k^2 \) when this matrix is singular.

Step 1: Condition for a matrix to be singular

A matrix is singular if and only if its determinant is zero. Therefore, we first need to find the determinant of matrix \( A \). The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by: \[ \det(A) = ad - bc \] For the matrix \( A = \begin{bmatrix} 2 - k & 2 \\ 1 & 3 - k \end{bmatrix} \), we have: - \( a = 2 - k \) - \( b = 2 \) - \( c = 1 \) - \( d = 3 - k \) The determinant of \( A \) is: \[ \det(A) = (2 - k)(3 - k) - (2)(1) \]

Step 2: Simplify the determinant expression

First, expand the terms: \[ \det(A) = (2 - k)(3 - k) - 2 \] Expanding \( (2 - k)(3 - k) \): \[ (2 - k)(3 - k) = 6 - 2k - 3k + k^2 = 6 - 5k + k^2 \] So, the determinant becomes: \[ \det(A) = 6 - 5k + k^2 - 2 = k^2 - 5k + 4 \]

Step 3: Set the determinant equal to zero

For the matrix to be singular, the determinant must be zero: \[ k^2 - 5k + 4 = 0 \]

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula: \[ k = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(4)}}{2(1)} \] Simplifying: \[ k = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm \sqrt{9}}{2} = \frac{5 \pm 3}{2} \] This gives two possible solutions for \( k \): \[ k = \frac{5 + 3}{2} = 4 \quad \text{or} \quad k = \frac{5 - 3}{2} = 1 \]

Step 5: Calculate \( 5k - k^2 \)

We now calculate \( 5k - k^2 \) for both values of \( k \): - For \( k = 4 \): \[ 5k - k^2 = 5(4) - (4)^2 = 20 - 16 = 4 \] - For \( k = 1 \): \[ 5k - k^2 = 5(1) - (1)^2 = 5 - 1 = 4 \]

Final Answer:

In both cases, \( 5k - k^2 = 4 \). Therefore, the value of \( 5k - k^2 \) is \( \boxed{4} \).