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 (Alternate Method):
Instead of constructing the matrix and calculating, use the scalar multiplication property of determinants: When a matrix of order n is multiplied by scalar k, its determinant gets multiplied by k^n.

Step 3: Detailed Explanation:
We are given: Matrix A is of order n=2 and det(A)=5.
We need det(2A). Scalar k=2.
Using the property det(kA) = k^n × det(A):
det(2A) = 2^2 × 5
det(2A) = 4 × 5
det(2A) = 20
This avoids expanding the 2x2 matrix and multiplying each element.

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