GATE CS 2016 Set 2 Q06 — Eigenvalues and Eigenvectors, LU Decomposition

We are given that the eigenvalues of the matrix $A$ are $\lambda_1 = 1$, $\lambda_2 = 2$, and $\lambda_3 = 4$.

Using the properties of determinants and eigenvalues:

1. Determinant of $A$: The determinant of a matrix is equal to the product of its eigenvalues. $$\det(A) = \lambda_1 \times \lambda_2 \times \lambda_3 = 1 \times 2 \times 4 = 8$$
2. Determinant of Transpose: The determinant of the transpose of a matrix is equal to the determinant of the matrix itself. $$\det\left(\left(A^{-1}\right)^T\right) = \det(A^{-1})$$
3. Determinant of Inverse: The determinant of the inverse of a matrix is the reciprocal of its determinant. $$\det(A^{-1}) = \frac{1}{\det(A)}$$

Combining these properties, we get: