1.If $A$ is a square matrix of order $3$ and $|A| = 4$, find the value of $|adj(2A)|$.
2.Evaluate $\begin{vmatrix} 1 & 1 & 1 \\ ^nC_1 & ^{n+1}C_1 & ^{n+2}C_1 \\ ^nC_2 & ^{n+1}C_2 & ^{n+2}C_2 \end{vmatrix}$.
3.Let $\Delta(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$. Find the exact value of $\Delta'(x)$.
4.If $A$ is a skew-symmetric matrix of order $3$, find the value of $|I + A| - |I - A|$.
5.Find the maximum value of a $3 \times 3$ determinant whose elements belong to the set $\{-1, 1\}$.
6.The system of linear equations $x + ky + 3z = 0$, $3x + ky - 2z = 0$, $2x + 3y - 4z = 0$ has a non-trivial solution. Find the value of $k$.
7.If $\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix} = k \cdot a^2 b^2 c^2$, find the integer $k$.
8.Given $A$ is an orthogonal matrix ($A A^T = I$). Find the possible values of $|A|$.
9.If $A^2 - A + I = O$, prove that $A^{-1} = I - A$.
10.Find the value of $x$ for which the matrix $A = \begin{bmatrix} 2 & x-1 \\ 3 & x \end{bmatrix}$ is its own inverse (i.e., $A^2 = I$).
11.Let $f(x) = \begin{vmatrix} \sec x & \cos x & \sec^2 x + \cot x\csc x \\ \cos^2 x & \cos^2 x & \csc^2 x \\ 1 & \cos^2 x & \cos^2 x \end{vmatrix}$. Evaluate $\int_0^{\pi/2} f(x) dx$.
12.If $A, B, C$ are the angles of a triangle, evaluate $\begin{vmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix}$.
13.Let $P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $Q = [q_{ij}]$ be two $3 \times 3$ matrices such that $Q - P^5 = I_3$. Find the value of $q_{21} + q_{31}$.
14.If $x, y, z$ are all different and non-zero, and $\begin{vmatrix} x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3 \end{vmatrix} = 0$, prove that $xyz = -1$.
15.If the system of equations $x + ay = 0$, $az + y = 0$, and $ax + z = 0$ has infinite solutions, find the value of $a$.
16.Evaluate $\begin{vmatrix} (a^x + a^{-x})^2 & (a^x - a^{-x})^2 & 1 \\ (b^x + b^{-x})^2 & (b^x - b^{-x})^2 & 1 \\ (c^x + c^{-x})^2 & (c^x - c^{-x})^2 & 1 \end{vmatrix}$.
17.If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $f(x) = \frac{1+x}{1-x}$, find the matrix $f(A)$.
18.Find the sum of all real roots of the equation $\begin{vmatrix} x-2 & 2x-3 & 3x-4 \\ x-4 & 2x-9 & 3x-16 \\ x-8 & 2x-27 & 3x-64 \end{vmatrix} = 0$.
19.Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = O$. Find the determinant of $A$.
20.Determine the values of $\theta \in [0, 2\pi]$ for which $\begin{vmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{vmatrix} = 3$.
21.If $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$ and $B = A^{-1}$, find the value of the element $b_{23}$ of matrix $B$.
22.Evaluate $\lim_{x \to 0} \frac{1}{x^3} \begin{vmatrix} 1 & \cos x & \sin x \\ \cos x & 1 & \sin 2x \\ \sin x & \sin 2x & 1 \end{vmatrix}$.
23.If $adj(A) = \begin{bmatrix} 2 & 1 & 3 \\ -1 & 0 & 2 \\ 3 & 1 & -1 \end{bmatrix}$, find the determinant of $A$.
24.Consider the system of equations $2x_1 - x_2 + 3x_3 = 1$, $3x_1 - 2x_2 + 5x_3 = 2$, $-x_1 + 4x_2 + x_3 = 3$. Find the sum of $x_1, x_2, x_3$.
25.Let $A$ be an involutory matrix ($A^2 = I$). Show that $|A| = \pm 1$. Furthermore, if $A \neq I$, show that $|I - A| \neq 0$ is impossible unless order is even.
26.Evaluate the determinant $\begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ bc & ca & ab \end{vmatrix}$.
27.If $a, b, c$ are the roots of $x^3 - px^2 + qx - r = 0$, evaluate $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ in terms of $p, q, r$.
28.Let $A = [a_{ij}]_{3 \times 3}$ where $a_{ij} = 2^{i-j}$. Evaluate $|A|$.
29.Find the condition for the lines $ax+by+c=0$, $bx+cy+a=0$, and $cx+ay+b=0$ to be concurrent, assuming $a+b+c \neq 0$.
30.Evaluate $\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$ completely.
31.Let $f(x) = \begin{vmatrix} x+\lambda & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda \end{vmatrix}$. Prove that $f(x) = \lambda^2(3x+\lambda)$.
32.If the matrix $A = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$ is orthogonal and $a, b, c$ are positive real numbers, find the value of $abc$.
33.Determine the area of the polygon formed by the solutions of $|x| + |y| = a$ using determinants.
34.If $\Delta = \begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix} = 0$, where $A, B, C$ are angles of a triangle, what type of triangle is it?
35.Let $A$ be a non-singular matrix. Prove that $adj(A^{-1}) = (adj A)^{-1}$.
36.Evaluate the determinant $\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}$.
37.Find the value of $x, y, z$ if $\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$.
38.Let $P$ and $Q$ be $3 \times 3$ matrices such that $P \neq Q$, $P^3 = Q^3$, and $P^2Q = Q^2P$. Evaluate $|P^2 + Q^2|$.
39.Find the maximum value of $f(x) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1 & 1 & 1+\cos x \end{vmatrix}$ for $x \in \mathbb{R}$.
40.If $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} = 0$ has $x = -9$ as a root, find the other two roots.
41.Given $A$ and $B$ are symmetric matrices of the same order. Prove that $AB - BA$ is a skew-symmetric matrix. What can you say about $|AB - BA|$ if the order is 3?
42.Solve the system: $\sin x + \cos y + \tan z = 2$, $\sin x - \cos y + \tan z = 0$, $2\sin x + 3\cos y - \tan z = 4$ for basic principal values.
43.Let $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$. Find the value of $\lim_{n \to \infty} \frac{1}{n} A^n$ if $\theta = \pi/2$.
44.If $\alpha, \beta, \gamma$ are the roots of $x^3 + px + q = 0$, find the value of $\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix}$.
45.If $A$ is an idempotent matrix ($A^2 = I$), evaluate $|A + I|$ assuming $A \neq -I$ and $A \neq I$, and order is $n$.
46.For a $3 \times 3$ matrix $A$, if $adj(2A) = k \cdot adj(A)$, find the value of $k$.
47.Evaluate $\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix}$.
48.Find the area of the region bounded by $3x+4y=12, 3x-4y=12, -3x+4y=12, -3x-4y=12$ using determinant formulas of constituent triangles.
49.Let $A$ be a $3 \times 3$ matrix such that $A \cdot adj(A) = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$. Find $|adj(adj A)|$.
50.If the system of equations $x + \lambda y - z = 0, \lambda x - y - z = 0, x + y - \lambda z = 0$ has a non-trivial solution, find the possible values of $\lambda$.
51.Evaluate $\begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{vmatrix}$.
52.Consider the matrix $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$. Show that $A^3 = I$. Hence find $A^{-1}$.
53.If $\begin{vmatrix} y+z & x & x \\ y & z+x & y \\ z & z & x+y \end{vmatrix} = k \cdot xyz$, find $k$.
54.Determine whether the following is true or false: The inverse of a symmetric non-singular matrix is always symmetric. Prove your claim.
55.Let $P$ be a $3 \times 3$ matrix such that $P^T = 2P + I$. Find $|P|$.
56.Find the value of $\begin{vmatrix} ^mC_1 & ^mC_2 & ^mC_3 \\ ^nC_1 & ^nC_2 & ^nC_3 \\ ^pC_1 & ^pC_2 & ^pC_3 \end{vmatrix}$ if $m=1, n=2, p=3$.
57.Solve for $x$: $\begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix} = 0$.
58.Find the equation of the plane passing through $(1, 2, -1)$, $(2, 3, 1)$ and $(3, -1, 2)$ using determinants.
59.Evaluate $\begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ac & bc & c^2+1 \end{vmatrix}$.
60.Let $A = \begin{bmatrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \end{bmatrix}$. Find the values of $a, b, c$ such that $A^T A = I$.
