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 equal to:
If \( A = \begin{bmatrix} 2-k & 2 \\ 1 & 3-k \end{bmatrix} \) is a singular matrix, then the value of \( 5k - k^2 \) is equal to:
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A \(2 \times 2\) matrix is singular when \(ad - bc = 0\). This condition implies linear dependence between rows or columns.
Updated On: May 6, 2026
- \( 0 \)
- \( 6 \)
- \( -6 \)
- \( -4 \)
- \( 4 \)
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Solution and Explanation
Concept:
A matrix is singular if its determinant is zero (\( |A| = 0 \)).
Step 1: Calculate the determinant.
Given matrix: \[ A = \begin{pmatrix} 2 - k & 1 \\ 2 & 3 - k \end{pmatrix} \] For a \(2 \times 2\) matrix, determinant is: \[ |A| = (2 - k)(3 - k) - (1)(2) \] Now expand: \[ (2 - k)(3 - k) = 6 - 2k - 3k + k^2 \] So, \[ |A| = 6 - 5k + k^2 - 2 \] \[ |A| = k^2 - 5k + 4 \]
Step 2: Use the singular condition.
Since the matrix is singular: \[ |A| = 0 \] So, \[ k^2 - 5k + 4 = 0 \] Rearrange: \[ k^2 - 5k = -4 \]
Step 3: Find the required expression.
We need: \[ 5k - k^2 \] Multiply the equation \(k^2 - 5k = -4\) by \(-1\): \[ -(k^2 - 5k) = -(-4) \] \[ 5k - k^2 = 4 \] Thus, \[ 5k - k^2 = 4 \] Final Answer: \[ \boxed{4} \]
Step 1: Calculate the determinant.
Given matrix: \[ A = \begin{pmatrix} 2 - k & 1 \\ 2 & 3 - k \end{pmatrix} \] For a \(2 \times 2\) matrix, determinant is: \[ |A| = (2 - k)(3 - k) - (1)(2) \] Now expand: \[ (2 - k)(3 - k) = 6 - 2k - 3k + k^2 \] So, \[ |A| = 6 - 5k + k^2 - 2 \] \[ |A| = k^2 - 5k + 4 \]
Step 2: Use the singular condition.
Since the matrix is singular: \[ |A| = 0 \] So, \[ k^2 - 5k + 4 = 0 \] Rearrange: \[ k^2 - 5k = -4 \]
Step 3: Find the required expression.
We need: \[ 5k - k^2 \] Multiply the equation \(k^2 - 5k = -4\) by \(-1\): \[ -(k^2 - 5k) = -(-4) \] \[ 5k - k^2 = 4 \] Thus, \[ 5k - k^2 = 4 \] Final Answer: \[ \boxed{4} \]
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