Question

If \[ A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} \] and the determinant of \(A\) is \(5\), then determinant of the matrix \(2A\) is

• A. \(10\)
• B. \(20\)
• C. \(5\)
• D. \(25\)

Hint:

For an \(n \times n\) matrix, multiplying the matrix by \(k\) multiplies the determinant by \(k^n\).

Answer 1

The Correct Option is B

Concept:
If \(A\) is an \(n \times n\) matrix, then: \[ |kA|=k^n|A| \] where \(k\) is a scalar and \(n\) is the order of the matrix.

Step 1: The given matrix is: \[ A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} \] So, \(A\) is a \(2 \times 2\) matrix.

Step 2: Given: \[ |A|=5 \]

Step 3: We need to find: \[ |2A| \]

Step 4: Since the matrix is of order \(2\), use the formula: \[ |kA|=k^2|A| \]

Step 5: Put \(k=2\): \[ |2A|=2^2|A| \] \[ |2A|=4 \times 5 \] \[ |2A|=20 \] \[ \boxed{20} \]