Matrices - Class 12

Let $y = f(x)$ be a twice differentiable, strictly decreasing function on $\mathbb{R}$. For all $x \in \mathbb{R}$, the rows of the matrix $M = \begin{pmatrix} y'' & e^x & 1 \\ |y'| & 1 & 0 \\ y & x & 1 \end{pmatrix}$ are linearly dependent. If $m$ and $n$ are the order and degree of the differential equation satisfied by $y = f(x)$ respectively, then find the value of $2m + 3n$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors representing the position vectors of points $A, B, C$ with respect to an origin $O$. A new origin $D$ is chosen with position vector $\vec{d} = x\vec{a} + y\vec{b} + z\vec{c}$ such that $x+y+z \neq 1$. Let $\vec{r}_A, \vec{r}_B, \vec{r}_C$ be the position vectors of $A, B, C$ with respect to $D$. A $3 \times 3$ matrix $M$ is formed such that its rows are the scalar components of the vectors $\vec{u}, \vec{v}, \vec{w}$ respectively, where: $\vec{u} = 7(\vec{r}_A \times \vec{r}_B) + 3(\vec{r}_B \times \vec{r}_C)$ $\vec{v} = 7(\vec{r}_B \times \vec{r}_C) + 7(\vec{r}_C \times \vec{r}_A)$ $\vec{w} = 3(\vec{r}_C \times \vec{r}_A) + \lambda(\vec{r}_B \times \vec{r}_A)$ If $M$ is a singular matrix for all such choices of $D$, then the value of $\lambda$ is:

Let $A = \begin{pmatrix} \text{sgn}(x^2 - x - 2) + 1 & \cos^{-1}(x) \\ \sin(2\pi x) & \text{sgn}(x + 1) \end{pmatrix}$, where $\text{sgn}(\cdot)$ denotes the signum function. If $S$ is the set of all real values of $x$ for which $A$ is an idempotent matrix, then the value of $4 \sum_{x \in S} x^2$ is equal to

A variable plane $P$ has intercepts $a(t), b(t),$ and $c(t)$ on the $x, y,$ and $z$ axes respectively, where $a(t) = t^3 + k$, $b(t) = \frac{15}{2}t^2$, and $c(t) = t^3 + 36t$ for a parameter $t \in [0, 4]$. Consider the system of planes given by the equations: $P_1: x + y + z = 1$ $P_2: x + 2y + 3z = 1$ $P_3: a(t)x + b(t)y + c(t)z = 1$ If the coefficient matrix of this system is non-singular for all $t \in [0, 4]$, what is the complete set of values for the real constant $k$?

Let $A = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix}$. Consider the system of linear equations given by $(A^{2024} - I) \mathbf{x} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $I$ is the $3 \times 3$ identity matrix and $\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$. If this system has infinitely many solutions, which of the following conditions must be true?

Let $A$ be a $3 \times 3$ matrix whose rows are given by the vectors $\vec{r}_1, \vec{r}_2, \vec{r}_3 \in \mathbb{R}^3$. It is given that $A$ is an involutory matrix and $\vec{r}_1, \vec{r}_2, \vec{r}_3$ are unit vectors. If $\vec{r}_1 = \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \frac{1}{\sqrt{3}}\hat{k}$ and $\vec{r}_2 = x \hat{i} + y \hat{j} + z \hat{k}$, then the maximum possible value of $y$ is:

Let $A(x) = \begin{pmatrix} 0 & x-1 & 0 \\ 0 & 0 & x \\ (x-1)^2 & 0 & 0 \end{pmatrix}$. A curve $C: y = f(x)$ passes through the point $(2, 1)$. The slope of the tangent to $C$ at any point $(x, f(x))$ is given by $\frac{1}{3}\text{tr}(A(x)^3)$. Let $S_1$ be the set of all $x$-coordinates where $f(x)$ attains a local minimum, and $S_2$ be the set of all $x$-coordinates where $f(x)$ attains a local maximum. Find the value of $20 \left( \sum_{x \in S_1} f(x) - \sum_{x \in S_2} f(x) \right)$. (If a set is empty, its sum is taken as $0$).

Let $S$ be the set of all real numbers $k$ for which the quadratic equations $x^2 - 2x + 1 - k = 0$ and $x^2 - 2\sqrt{k}x + k - 1 = 0$ have at least one common real root. Let $A$ be a $2 \times 2$ matrix with real entries such that its trace is $k^2 - 4k + 5$ and its determinant is $k-2$. If $k \in S$ is chosen such that the trace of $A$ is minimized, which of the following is equal to $A^{2024}$?

Let $A = \begin{pmatrix} 3 & -6 \\ 1 & -2 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 2 \\ -2 & 6 \end{pmatrix}$. Let $Y(t)$ be a $2 \times 2$ matrix-valued function satisfying the differential equation $\frac{dY}{dt} = AY - YA$ with the initial condition $Y(0) = B$. Find the sum of the elements of the matrix $Y(\ln 2)$.

Let $A(x) = \begin{pmatrix} x & [x] & \{x\} \\ [x] & \{x\} & x \\ \{x\} & x & [x] \end{pmatrix}$, where $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}$ denotes the fractional part of $x$. Let $P$ be the $3 \times 3$ matrix obtained by interchanging the first and second rows of the identity matrix $I_3$. If $B(x) = \text{adj}(P) A(x)^T P$ and $f(x)$ is the scalar triple product of the vectors formed by the first, second, and third rows of $B(x)$ respectively, find the value of $16 \int_{-1}^{2} f(x) dx$.

Let $P$ be a $3 \times 3$ matrix with complex entries satisfying $P P^\theta = 2I$ and $\det(P) = 2\sqrt{2} i$, where $P^\theta$ is the conjugate transpose of $P$. Let $Q$ be a $3 \times 3$ matrix given by $Q = \begin{pmatrix} e^{i\pi/3} & 0 & 0 \\ 0 & e^{-i\pi/3} & 0 \\ 0 & 0 & 1 \end{pmatrix}$ Define $M = P Q P^T$ and $N = P Q P^\theta$, where $P^T$ is the transpose of $P$. If $Z = \frac{\det(M)}{2 \det(N)} N$, find the value of $\sum_{k=1}^{5} \text{Tr}(Z^k)$

Let $A(x)$ and $B(x)$ be $3 \times 3$ matrices defined for $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ as follows: $A(x) = \begin{bmatrix} x + x^2 \sin\left(\frac{1}{x}\right) & \sin x & \cos x \\ 0 & \cos x & \sin x \\ 0 & -\sin x & \cos x \end{bmatrix} \text{ for } x \neq 0, \text{ and } A(0) = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ $B(x) = \begin{bmatrix} \sec x & \tan x & 0 \\ \tan x & \sec x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ If $M(x) = (A(x) B(x))^T$, then the value of $\lim_{x \to 0} \frac{\det(M(x))}{e^{2x} - 1}$ is:

Let $t_n$ be a geometric progression of real numbers with first term $t_1 = 1$ and a non-zero common ratio $r$. For each integer $k \ge 1$, define the $3 \times 3$ matrix $M_k$ as: $M_k = \begin{pmatrix} t_k & t_{k+1} & t_{k+2} \\ -t_{k+1} & t_k & t_{k+1} \\ -t_{k+2} & -t_{k+1} & t_k \end{pmatrix}$ If $r$ satisfies the inequality $\det( - (M_1 M_2)^T ) \ge -16 t_4$, find the sum of all possible integer values of $r$ in the interval $[-3, 3]$.

Let $S = \{ z \in \mathbb{C} : |z| \le 3 \}$. For any $z \in S$, a matrix $A$ is defined as: $A = \begin{pmatrix} [|z|] & -1 & 1 \\ 1 & [|z-3|] & -1 \\ -1 & 1 & [|z-4i|] \end{pmatrix}$ where $[x]$ denotes the greatest integer less than or equal to $x$. Let $V$ be the volume of the parallelepiped whose coterminous edges are the column vectors of the matrix $\text{Adj}(A)$. Find the maximum possible value of $V$.

Let A(x) = \\begin{pmatrix} 2 & 2x & 2x^2 \\\\ 2 & 2x-2 & 2x^2-4x+2 \\\\ 2 & 2[x] & 2[x]^2 \\end{pmatrix}, where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f(x) = \\sqrt{\\det(\\text{adj}(A(x)))}$. Define the sets $S_1$ and $S_2$ in the interval $(0, 10)$ as:\nS_1 = \\{x \\in (0, 10) \\mid f(x) \\text{ is discontinuous}\\}\\nS_2 = \\{x \\in (0, 10) \\mid f(x) \\text{ is non-differentiable}\\}\\nIf $M$ is the maximum value of $f(x)$ on $\\mathbb{R}$, find the value of $10M + n(S_1) + n(S_2)$, where $n(S)$ denotes the number of elements in set $S$.

Let $w, x, y, z \in \mathbb{R}$ such that the following matrix equation holds: $\begin{pmatrix} w & x \\ y & z \end{pmatrix} = \begin{pmatrix} \int_{-e}^{-1} t \ln|t| dt & \int_{-2}^{2} \frac{t^3}{\sqrt{4-t^2}} dt \\ \int_{1}^{e} t \ln t dt & \int_{1}^{2} \left( 3t^2 - 2t + \frac{5}{4} \right) dt \end{pmatrix}$ Find the value of $w + x + y + z$.

Let $A(x) = \begin{pmatrix} e^{1/x} & \sin x & 1 \\ \frac{1}{x} & 1 & \frac{1}{x} \\ 1 & x - x^2 \ln\left(1+\frac{1}{x}\right) & 1 \end{pmatrix}$ Evaluate $\lim_{x \to \infty} \left( x \det(A(x)) \right)^{x^2}$

Let $\vec{u}$ and $\vec{v}$ be two unit vectors in $\mathbb{R}^3$ such that the angle between them is $\frac{2\pi}{3}$. Let $L_1$ and $L_2$ be the lines passing through the origin parallel to $\vec{u}$ and $\vec{v}$ respectively. Let $\vec{w}$ be a unit vector such that the angle between the line $L_3$ (parallel to $\vec{w}$) and $L_1$ is $\frac{\pi}{3}$, and the angle between $L_3$ and $L_2$ is also $\frac{\pi}{3}$. If the algebraic projection of $\vec{w}$ on $\vec{u}$ is positive and the algebraic projection of $\vec{w}$ on $\vec{v}$ is negative, let $M$ be the $3 \times 3$ matrix whose rows are the components of $\vec{u}, \vec{v},$ and $\vec{w}$ in a standard orthonormal basis. The value of $|\det(M)|$ is:

Let $y = f(x)$ be the particular solution to the exact differential equation $(2x y e^{x^2})dx + (e^{x^2} + 2y)dy = 0$ satisfying $f(0) = 1$. Let the function $g(x)$ be defined as $g(x) = \lim_{n \to \infty} \frac{246 x^{2n} + \frac{1}{246}}{x^{2n} + 1}$ Define the matrix $A(x)$ as $A(x) = \begin{pmatrix} f(x) & f(g(x)) \\ e^{x^2} & e^{(g(x))^2} \end{pmatrix}$ The product of all real values of $x$ for which the matrix $A(x)$ is singular is:

Let the involutory matrix $A = \frac{1}{3} \begin{pmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{pmatrix}$. A non-zero complex vector $\vec{Z} = \begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix}$ satisfies the transformation equation $A\vec{Z} = -\vec{Z}$. If the moduli of its components are given by $|z_1| = 7$, $|z_2| = 8$, and $|z_3| = 15$, find the value of $|z_1^2 + z_2^2 + z_3^2|$.

Let $f(x) = |x| + x^2$ for $x \in [-1, 1]$. Consider the matrix $M(x) = \begin{pmatrix} f'(x) & 2 & 2 \\ 1 & -1 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ defined for all $x \in (-1, 1) \setminus \{0\}$. Let $C$ be the set of all $c \in (-1, 1) \setminus \{0\}$ such that $M(c)$ is a singular matrix. If $C$ is non-empty, let $S = \sum_{c \in C} c$. If $C$ is empty, let $S = 10$. Find the value of $S$.

Let $A = I + K$ be a $4 \times 4$ matrix, where $I$ is the identity matrix and $K = [k_{ij}]$ is a skew-symmetric matrix with real entries. It is given that $k_{12}=1$, $k_{13}=2$, $k_{14}=3$, $k_{24}=2$, $k_{34}=1$, and $k_{23}=-x$ for some $x > 0$. If $\det(K) = 36$, find the value of $\det(A)$.

Let $l, m, n$ be the direction cosines of a line in three-dimensional space. Define the matrix $K$ as $K = \begin{pmatrix} 0 & -n & m \\ n & 0 & -l \\ -m & l & 0 \end{pmatrix}$ Let $M = 5I + \sqrt{11}K + 3K^2$, where $I$ is the $3 \times 3$ identity matrix. Find the value of $\det(M)$.

Let $S = (\lambda^2 - 1)I_3$ and $K$ be a $3 \times 3$ skew-symmetric matrix with $K_{12} = K_{13} = K_{23} = 1$. Let $A = S + K$ and $B = \begin{pmatrix} \lambda^2 + \lambda - 2 \\ \lambda^2 + |\lambda| - 2 \\ \lambda - |\lambda| \end{pmatrix}$. If $S_1$ is the set of all real values of $\lambda$ for which the system of equations $AX = B$ has infinitely many solutions, and $S_2$ is the set of all real values of $\lambda$ for which the system has no solution, then find $S_1$ and $S_2$.

Let $M$ be a $50 \times 50$ matrix whose entries are given by $M_{i,j} = \sum_{k=1}^{\min(i,j)} \frac{(2k)!}{k!} \binom{i}{k} \binom{j}{k}$ for $1 \le i, j \le 50$. Find the maximum integer $E$ such that $3^E$ divides $\det(M)$.

Let $m$ and $n$ be the order and degree, respectively, of the differential equation $\frac{d^3y}{dx^3} = \left( 1 + \left(\frac{d^2y}{dx^2}\right)^4 \right)^{\frac{1}{3}}$ Let $A$ be an $m \times m$ matrix such that $A^m = O$ and $A^{m-1} \neq O$, where $O$ is the null matrix. If $M = (I + A)^{-n} + (I - A)^{-n}$, then the value of $\det(n M)$ is

Let $M(x) = \begin{pmatrix} \int_{0}^{x} \frac{t^2 2^t}{2^t + 2^{x-t}} [t] dt & x^2 \sin x \\ e^{\cos x} & \int_{0}^{x} \frac{(x-t)^2 2^t}{2^t + 2^{x-t}} [x-t] dt \end{pmatrix}$, where $[ \cdot ]$ denotes the greatest integer function.

If $F(x) = \text{Tr}(M(x))$, find the value of: $\lim_{h \to 0^+} \frac{F(3+h) - F(3-h)}{h}$

Let $A, B,$ and $C$ be $3 \times 3$ real skew-symmetric matrices. It is given that $\text{Tr}(A^2) = -2$, $\text{Tr}(B^2) = -8$, $\text{Tr}(C^2) = -18$, and $\text{Tr}(AB) = \text{Tr}(BC) = \text{Tr}(CA) = 0$. For any vector $\vec{x} \in \mathbb{R}^3$, the matrices act as cross products: $A\vec{x} = \vec{a} \times \vec{x}$, $B\vec{x} = \vec{b} \times \vec{x}$, and $C\vec{x} = \vec{c} \times \vec{x}$, where $\vec{a}, \vec{b}, \vec{c}$ form a right-handed orthogonal system (i.e., $[\vec{a} \vec{b} \vec{c}] > 0$). If $P = AB - BA$, $Q = BC - CB$, and $R = CA - AC$, find the value of $\text{Tr}(PQR)$.

Let $A_k = \begin{pmatrix} \frac{1}{k(k+1)} & 2k-1 \\ -(2k-1) & 0 \end{pmatrix}$ for $k \in \mathbb{N}$. Let $M_n = \sum_{k=1}^n A_k$. If $S$ is the set of all natural numbers $n$ such that the matrix $M_n$ has real eigenvalues, find the number of elements in $S$.

Let $P(x) = x^3 - 4x^2 + 3x + 2$. Let $r_1, r_2, r_3$ be the roots of $P(x)=0$ such that $r_1$ and $r_2$ are irrational and $r_3$ is rational. Consider the system of linear equations in $x, y, z$: $x + y + z = 1$ $r_1 x + r_2 y + r_3 z = 2$ $r_1^2 x + r_2^2 y + [\lambda] z = 3$ where $[\cdot]$ denotes the greatest integer function. Let $S$ be the set of all real values of $\lambda$ for which the system has no solution. Then $S$ is equal to:

Let $A(x)$ be a $3 \times 3$ orthogonal matrix with continuously differentiable entries for all $x \in \mathbb{R}$. Let $M(x)$ be a $3 \times 3$ matrix with differentiable entries satisfying the differential equation $M'(x) = \left( \int_{x}^{x^2} A(t)^T A'(t) dt \right) M(x)$ for all $x \in \mathbb{R}$, with $M(1) = A(1)$. Find the value of $\frac{d}{dx} \left( \det(M(x)) \right)$ at $x = 2024$.

Let $A = \begin{pmatrix} 0 & 3 & 4 \\ 0 & 0 & 5 \\ 0 & 0 & 0 \end{pmatrix}$ and $I$ be the $3 \times 3$ identity matrix. Consider the infinite series of matrices $S = \sum_{k=1}^{\infty} (k I - A)^{-1} A ((k+1)I - A)^{-1}$ Then the sum of all the elements of the matrix $S$ is:

Let $S$ be the set of all $3 \times 3$ matrices with entries from $\{-1, 1\}$. A matrix $A$ is chosen at random from $S$. Let $V$ be the volume of the parallelepiped formed by the column vectors of $A$. Let $E$ be the event that the sum of the roots of the characteristic equation of $A$ is $3$. Find the conditional probability $P(V = 0 \mid E)$.

Let $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 3 \end{pmatrix}$ and let $\lambda_1 < \lambda_2 < \lambda_3$ be the roots of its characteristic equation. Consider two planes $P_1$ and $P_2$ given in parametric form as: $P_1: \vec{r} = \hat{i} + s(-\lambda_1 \hat{i} + \lambda_2 \hat{j}) + t(\lambda_2 \hat{j} + \lambda_3 \hat{k})$ $P_2: \vec{r} = \hat{j} + p(\lambda_3 \hat{i} - \lambda_1 \hat{j}) + q(\lambda_3 \hat{j} + \lambda_2 \lambda_3 \hat{k})$ where $s, t, p, q \in \mathbb{R}$. The angle between the planes $P_1$ and $P_2$ is:

Let $M(\theta) = \begin{pmatrix} \sin 2\theta & \cos 2\theta & \sin \theta \\ \cos 2\theta & -\sin 2\theta & \cos \theta \\ \sin \theta & \cos \theta & \sin\theta + \cos\theta \end{pmatrix}$. Let $S$ be the set of all $\theta \in [0, 2\pi]$ such that the matrix $M(\theta)$ has at least one integer eigenvalue. Find the sum of all elements in $S$.

Let $A_k(\theta) = \frac{1}{\sqrt{2}} \begin{pmatrix} e^{2 i k \theta} & e^{i k \theta} \\ -e^{i k \theta} & 1 \end{pmatrix}$ for $k \in \mathbb{N}$ and $\theta \in [0, \pi]$. Let $M(\theta) = A_1(\theta) A_2(\theta) \dots A_{10}(\theta)$. If $\det(M(\theta)) + \det(M^{-1}(\theta)) = \sqrt{3}$, then the sum of all possible values of $\theta$ is:

Let the system of linear equations $x + y + z = 3$ $2x + 3y + z = 6$ $3x + 4y + \alpha z = \beta$ have infinitely many solutions. Let $S$ be the set of all solutions $(x, y, z)$ of this system that also satisfy the inequalities $x \ge 0$, $y \ge 0$, and $z \ge 0$. Let $M$ be the maximum value of the objective function $Z = 2x + 4y + 5z$ on $S$, $L$ be the square of the length of the line segment represented by $S$, and $N$ be the number of points in $S$ where all coordinates are integers. Which of the following is NOT a correct value?

Let $A(x)$ be a $3 \times 3$ symmetric orthogonal matrix with continuous entries for all $x \in \mathbb{R}$. The trace of $A(x)$ is $1$ and its first column is given by $C_1(x) = \begin{pmatrix} \cos 2x \\ -2\sin^2 x \cos x \\ -2\sin x \cos^2 x \end{pmatrix}$ If $B = \int_{-\pi/2}^{\pi/2} A(x) \, dx$, then the value of $\frac{9}{\pi} \det(B)$ is

Let $A$ be a $3 \times 3$ matrix with real entries such that $AA^T = I$, $A^2 = I$, and $\text{Tr}(A) = 1$. Let $\vec{r}_1, \vec{r}_2, \vec{r}_3$ be the vectors formed by the first, second, and third rows of $A$ respectively. Let $\vec{u}$ be a vector along the internal angle bisector of $\vec{r}_1$ and $\vec{r}_2$ with magnitude $2$. Let $\vec{v}$ be a vector along the internal angle bisector of $\vec{r}_2$ and $\vec{r}_3$ with magnitude $\sqrt{2}$. If $\vec{w} = \vec{u} \times \vec{v}$, then the value of $\sqrt{2} \vec{w} \cdot (3\vec{r}_1 - 4\vec{r}_2 + 5\vec{r}_3)$ is:

Let $S$ be the set of all points $(x, y) \in \mathbb{R}^2$ such that the matrix $A = \begin{pmatrix} \sin(\sin^{-1} x) & \cos(\sin^{-1} x) \\ \sin(\sin^{-1} y) & -\cos(\sin^{-1} y) \end{pmatrix}$ is an orthogonal matrix. Let $L$ be the total arc length of the locus $S$, and let $k = \frac{L}{\pi}$. If $M = \begin{pmatrix} k & 1 \\ -1 & k \end{pmatrix}$, find the value of $\det(M^3 - 2M^2 + 2M)$.

Let $A = \begin{pmatrix} x & 1 & 1 \\ 1 & x-2 & 1 \\ 1 & 1 & x-2 \end{pmatrix}$ and $B = \begin{pmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 3 & 1 & x-2 \end{pmatrix}$. Let $f(x) = \frac{|A|}{|B|}$. A sequence of matrices $M_n$ is constructed such that $|M_n| - |M_{n-1}| = (f(x))^n$ for $n \ge 2$. If the sequence of determinants $|M_n|$ converges to a finite limit as $n \to \infty$, find the sum of all possible integers $x \in [-5, 5]$.