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

Step 1: Understanding the Concept:
This question tests the property of determinants under scalar multiplication. When a matrix is multiplied by a scalar constant, its determinant is scaled by that constant raised to the power of the matrix's order (size).
Step 2: Key Formula or Approach:
For any square matrix A of order $n$ and any scalar $k$, the determinant of the matrix $kA$ is given by the formula: \[ \det(kA) = k^n \det(A) \] Step 3: Detailed Explanation:
We are given the following information:
The matrix A is a 2x2 matrix, so its order is $n=2$.
The determinant of A is $\det(A) = 5$.
We need to find the determinant of the matrix 2A, so the scalar is $k=2$.
Using the formula from Step 2: \[ \det(2A) = 2^n \det(A) \] Substitute the values of $n$ and $\det(A)$: \[ \det(2A) = 2^2 \times 5 \] \[ \det(2A) = 4 \times 5 \] \[ \det(2A) = 20 \] Step 4: Final Answer:
The determinant of the matrix 2A is 20. Therefore, option (B) is the correct answer.