Mock Test 2 — Matrices

If $A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$, then the power matrix $A^n$ simplifies using induction to:

If $A$ is an idempotent matrix ($A^2 = A$), then the matrix expression $B = I - A$ satisfies which property?

The maximum possible rank for a completely non-zero skew-symmetric matrix of order 3 is:

If the matrix product $AB$ is well-defined and evaluates to an identity matrix ($AB = I$), then we can conclude that:

If $A$ and $B$ are two invertible square matrices of the same order, the inverse of their product $(AB)^{-1}$ is given by:

The system of linear equations $x + 2y + 3z = 1$, $2x + 4y + 6z = 2$, $3x + 6y + 9z = 3$ has:

If $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$, then the power matrix $A^n$ is equal to:

If the trace of a square matrix expression satisfies $\text{tr}(A^T A) = 0$ for a real matrix $A$, then matrix $A$ must be:

If $A$ is an orthogonal matrix, its transpose $A^T$ must satisfy which condition?

The linear system $AX = B$ has a unique solution if the ranks of the matrices satisfy:

Find the value of $x$ for which the matrix $A = \begin{bmatrix} x-1 & 2 \\ 3 & x+4 \end{bmatrix}$ becomes singular ($|A| = 0$) and has a positive integer root.

If $A$ is an involutory matrix ($A^2 = I$), evaluate the value of the determinant $|A^4|$.

Determine the rank of the augmented matrix $[A \mid B]$ for a system that has no solution, if the rank of the coefficient matrix is $\text{rank}(A) = 2$.

Assertion (A): The product of two symmetric matrices $A$ and $B$ is symmetric if and only if they commute ($AB = BA$).
Reason (R): Taking the transpose of a product gives $(AB)^T = B^T A^T$. If they are symmetric, this becomes $BA$, which equals $AB$ only if they commute.

Assertion (A): An odd-order skew-symmetric matrix can never be inverted.
Reason (R): The determinant of any odd-order skew-symmetric matrix is identically zero, making the matrix singular.