1.Answer: $-7$. (The determinant of a $1 \times 1$ matrix is simply the element itself).
2.Answer: $(2 \times 3) - (4 \times 1) = 6 - 4 = 2$.
3.Answer: $4x - 6 = 10 \implies 4x = 16 \implies x = 4$.
4.Answer: $(\sin \theta)(\sin \theta) - (-\cos \theta)(\cos \theta) = \sin^2 \theta + \cos^2 \theta = 1$.
5.Answer: $x(x) - (x+1)(x-1) = x^2 - (x^2 - 1) = 1$.
7.Answer: Since it is a diagonal matrix, $|A| = 1 \times 2 \times 3 = 6$.
8.Answer: No. Determinants are only defined for square matrices.
9.Answer: Expanding along row 1: $1(1\cdot1 - 3\cdot0) - 2(0 - 0) + 0 = 1$.
10.Answer: $(2\cdot1 - 4\cdot5) = (2x\cdot x - 4\cdot6) \implies 2 - 20 = 2x^2 - 24 \implies -18 = 2x^2 - 24 \implies 2x^2 = 6 \implies x^2 = 3 \implies x = \pm \sqrt{3}$.
11.Answer: The value remains unchanged.
12.Answer: $|A| = |A^T|$.
14.Answer: $0$ (Because Row 1 and Row 2 are identical).
15.Answer: $|2A| = 2^3 |A| = 8 \times 4 = 32$.
17.Answer: The sign is reversed (multiplied by $-1$).
19.Answer: False. (Only $|AB| = |A||B|$ is generally true).
20.Answer: $0$. (Take 10 common from Row 2, it becomes identical to Row 1).
21.Answer: $\Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$
23.Answer: $\frac{1}{2} |1(0-3) - 0 + 1(18-0)| = \frac{1}{2} |-3 + 18| = \frac{15}{2} = 7.5$ sq. units.
24.Answer: $\frac{1}{2} \begin{vmatrix} 2 & -6 & 1 \\ 5 & 4 & 1 \\ k & 4 & 1 \end{vmatrix} = \pm 35$.
25.Answer: It is the determinant obtained by deleting the $i$-th row and $j$-th column in which element $a_{ij}$ lies.
26.Answer: Delete row 1 and column 1. The remaining element is $5$. So, $M_{11} = 5$.
27.Answer: $A_{ij} = (-1)^{i+j} M_{ij}$.
28.Answer: The element is $a_{12}$. Its minor $M_{12} = 4$. Cofactor $A_{12} = (-1)^{1+2}(4) = -4$.
30.Answer: The value of the determinant ($\Delta$).
31.Answer: The cofactor matrix.
32.Answer: Interchange the diagonal elements ($a$ and $d$) and change the sign of the off-diagonal elements ($b$ and $c$). $adj(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
33.Answer: $|A|I$ (where $I$ is the identity matrix).
35.Answer: $A^{-1} = \frac{1}{|A|} adj(A)$.
36.Answer: No. Division by zero ($|A|=0$) is not defined.
37.Answer: $|adj(A)| = |A|^{n-1}$.
38.Answer: $|adj(A)| = 5^{3-1} = 5^2 = 25$.
39.Answer: False. The correct property is the reversal law: $(AB)^{-1} = B^{-1}A^{-1}$.
40.Answer: The Identity matrix itself ($I^{-1} = I$).
42.Answer: Unique solution.
43.Answer: infinitely many.
44.Answer: $\Delta$ must not be equal to zero ($\Delta \neq 0$).
45.Answer: Zero solution (or Trivial solution: $x=0, y=0, z=0$).
46.Answer: Non-trivial (infinitely many) solutions.
47.Answer: True. Parallel lines never intersect, so there is no solution.
48.Answer: Two. (Derivative of row 1 keeping row 2 constant + Derivative of row 2 keeping row 1 constant).
49.Answer: $\begin{vmatrix} f'(x) & g'(x) \\ a & b \end{vmatrix}$. (Since the second row is constant, its derivative is zero, so the second determinant vanishes).
50.Answer: First evaluate $\Delta = 1(1) - x(0) = 1$. The derivative of the constant $1$ is $0$.
