GATE CS 2016 Set 2 Q.16 (CS Q.6): Linear Algebra

Suppose that the eigenvalues of matrix $A$ are 1, 2, 4.

The determinant of $(A^{-1})^T$ is ________.

Your answer

Official answer and solution

Solution

Compute the determinant of matrix A from its eigenvalues, then apply determinant properties of matrix inverse and transpose.

The determinant of any matrix $A$ equals the product of its eigenvalues:

\det(A) = \lambda_1 \times \lambda_2 \times \lambda_3 = 1 \times 2 \times 4 = 8

For any invertible matrix $A$ :

\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{8} = 0.125

The determinant of a matrix equals the determinant of its transpose:

\det((A^{-1})^T) = \det(A^{-1}) = 0.125

Therefore, the determinant is 0.125 (or 1/8).

Final answer

0.125

Common trap

Inverting the eigenvalues after taking determinant instead of using $\det(A^{-1}) = 1/\det(A)$ , or thinking transpose changes the determinant.

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