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:

Remember that when a matrix is multiplied by a scalar \(k\), every element gets multiplied by \(k\). When calculating the determinant, each of the \(n\) rows (or columns) has a common factor of \(k\), which can be taken out. This results in the factor \(k\) being taken out \(n\) times, leading to the formula \(\det(kA) = k^n \det(A)\). This is a common source of error where students might mistakenly think \(\det(kA) = k \det(A)\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a 2x2 matrix A with its determinant equal to 5.
We need to find the determinant of the matrix 2A, which is obtained by multiplying every element of A by the scalar 2.

Step 2: Key Formula or Approach:
For a square matrix A of order \(n\) and a scalar \(k\), the determinant of the matrix \(kA\) is given by the property:
\[ \det(kA) = k^n \det(A) \]

Step 3: Detailed Explanation:
We are given the following information:
The matrix A is of order \(n=2\).
The determinant of A is \(\det(A) = 5\).
We need to find the determinant of the matrix 2A.
Here, the scalar \(k\) is 2.
Using the formula from Step 2:
\[ \det(2A) = 2^n \det(A) \]
Substitute the values \(n=2\) and \(\det(A)=5\):
\[ \det(2A) = 2^2 \times 5 \]
\[ \det(2A) = 4 \times 5 \]
\[ \det(2A) = 20 \]

Step 4: Final Answer:
The determinant of the matrix 2A is 20.