Question

If A is a non singular square matrix of order 3 such that \(A^3 = 4A^2\) then value of |A| is:

• A. 16
• B. 64
• C. 4
• D. 8

Answer 1

The Correct Option is B

To find the value of \(|A|\) for the matrix \(A\) given the equation \(A^3 = 4A^2\), we can follow these steps:

Since \(A\) is a non-singular square matrix, it implies that the determinant \(|A| \neq 0\). Given the equation:
\(A^3 = 4A^2\)

We can factor out \(A^2\) from both sides (since \(A^2\) is non-zero as A is non-singular):
\(A^2(A - 4I) = 0\)

Since \(A\) is non-singular, \(|A| \neq 0\), which implies that \(|A - 4I| = 0\). Here, \(I\) is the identity matrix of order 3.

This means that 4 is an eigenvalue of \(A\). If one eigenvalue of \(A\) is 4, and since the determinant \(|A|\) is the product of its eigenvalues, \(|A| = 4 \cdot 4 \cdot 4 = 4^3\).

Therefore,
|A| = 64

Thus, the value of \(|A|\) is 64.