Mock Test 1 — Determinants

If $A$ is a square matrix of order 3 and $|A| = -2$, then the value of the scaled determinant $|-2A|$ is:

The value of the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 5 & 8 & 9 \end{vmatrix}$ is equal to:

If $|\text{adj}(A)| = 25$ for a square matrix $A$ of order 3, then the possible values of the determinant $|A|$ are:

The system of equations $2x + 4y = 6$ and $3x + 6y = 9$ has:

The area of a triangle with vertices $(1,1)$, $(1,4)$, and $(5,1)$ evaluated using determinants is:

If three coordinate points $(1,3)$, $(2,5)$, and $(k,7)$ are collinear, the value of $k$ must be:

For a homogeneous system of linear equations $AX = 0$ with 3 unknowns, a non-trivial solution exists if and only if:

If the product of two determinants satisfies $|A||B| = 12$ and $|A| = -3$, then the value of $|B|$ is:

The characteristic equation of the matrix $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ is:

If a row of a determinant is multiplied by the cofactors of a different row, the sum of these products is always:

If $A$ is a square matrix of order 3 such that $|A| = 2$, evaluate the value of the determinant of its nested adjoint: $|\text{adj}(\text{adj}(A))|$.

Find the value of $x$ that satisfies the determinant equation: $\begin{vmatrix} x & 4 \\ 2 & x \end{vmatrix} = 0$ for a positive integer solution.

Determine the trace value of a matrix if its eigenvalues are $\lambda_1 = 4$ and $\lambda_2 = -1$.

Assertion (A): The system of equations $x + y = 3$ and $x + y = 5$ is inconsistent and has no solution.
Reason (R): For an inconsistent system, the base coefficient determinant satisfies $\Delta = 0$ and at least one of the modified determinants ($\Delta_x, \Delta_y$) is non-zero.

Assertion (A): The determinant of a matrix $A$ and its transpose $A^T$ are always equal ($|A| = |A^T|$).
Reason (R): Expanding a determinant along the rows yields the exact same numerical calculation values as expanding it along the columns.