Mock Test 1 — Matrices

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, then the product matrix $AB$ is:

If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix of the same order, then the product matrix $ABA$ is always:

The trace of the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is equal to:

If the rank of the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{bmatrix}$ is 2, then the value of $k$ can be:

If $A$ is an orthogonal matrix of order 3, then the value of its determinant $|A|$ must be:

The inverse matrix of $A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$ is:

If a square matrix satisfies $A^2 = A$, it is idempotent. The value of $(I-A)^2$ is:

The system of equations $x + y = 2$ and $2x + 2y = 5$ has:

If $A^T$ is the transpose of $A$, then the matrix product expression $(A^T)^T$ always simplifies to:

If $|A| = 4$ for a square matrix of order 3, the value of $|\text{adj}(A)|$ is:

Find the index of nilpotency for the matrix $A = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix}$.

If $A$ and $B$ are square matrices of order 3 such that $\text{tr}(AB) = 15$, find the value of $\text{tr}(BA)$.

Determine the rank of the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$.

Assertion (A): The system of homogeneous linear equations $AX = O$ always has at least one valid solution.
Reason (R): Setting all variables to zero ($X = O$) always satisfies any homogeneous linear system, producing the trivial solution.

Assertion (A): For any two square matrices $A$ and $B$, the expansion identity $(A+B)^2 = A^2 + 2AB + B^2$ is always true.
Reason (R): Matrix multiplication is non-commutative in general ($AB \neq BA$), so the middle terms expand to $AB + BA$ and cannot be merged into $2AB$.