Previous Year NEET Question- Problem Of Matrices

Let $A$ be a $3 \times 3$ matrix with eigenvalues $1, -1, 2$. If $|A| = -2$, then the determinant of $A^2$ is:

Answer: 16

Explanation:

The determinant of a matrix is the product of its eigenvalues. Since $A$ has eigenvalues $1, -1, 2$, its determinant is $1 \cdot (-1) \cdot 2 = -2$. The determinant of $A^2$ is the square of the determinant of $A$, so it is $(-2)^2 = 4$.

If $A$ is a $3 \times 3$ matrix such that $A^2 = A$, then $A$ is not necessarily singular.

Answer: False

Explanation:

A matrix is singular if its determinant is zero. If $A^2 = A$, then $A$ is not necessarily invertible, so its determinant may be zero.

2017:** If $A$ is a $3 \times 3$ matrix, then $A^2$ is a $3 \times 3$ matrix.