Question

If $A$ is a square matrix of order 3 and $|A|=5$, then the value of $|2A^{T}|$ is

• A. -10
• B. 10
• C. 40
• D. -40

Hint:

When a constant comes out of a determinant, it is raised to the power of the matrix order.

Answer 1

The Correct Option is C

To find the value of \(|2A^T|\), we need to understand certain properties of determinants and transposition of matrices.

1. Property 1: Transpose of a Matrix
   • The determinant of a matrix remains unchanged when the matrix is transposed. Therefore, \(|A^T| = |A|\).
2. Given Information
   • The determinant of the original matrix \(A\) is given as \(|A| = 5\).
3. Transposition Impact
   • From Property 1, we have \(|A^T| = |A| = 5\).
4. Property 2: Determinants and Scalar Multiplication
   • If a scalar \(k\) multiplies a square matrix \(A\) of order \(n\), the determinant of the resultant matrix is given by \(|kA| = k^n \cdot |A|\), where \(n\) is the order of the matrix.
   • In our case, since \(A\) is a 3x3 matrix, \(n = 3\).
5. Calculate |2A^T|
   • Using Property 2, we have \(|2A^T| = 2^3 \cdot |A^T|\).
   • Since \(|A^T| = |A| = 5\), substituting the values gives us:

\(|2A^T| = 2^3 \cdot 5 = 8 \cdot 5 = 40\)

Thus, the value of \(|2A^T|\) is 40.