Matrices and Determinants — JEE Main PYQs

If the system of equations

$$x + 5y + 6z = 4,$$ $$2x + 3y + 4z = 7,$$ $$x + 6y + az = b$$

has infinitely many solutions, then the point $(a, b)$ lies on the line

Consider the matrices $A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}$. If matrices P and Q are such that $PA = B$ and $AQ = B$, then the absolute value of the sum of the diagonal elements of $2(P + Q)$ is __________.

Let $S = \left\{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0 \right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in $S$, for which the sum of the diagonal elements is equal to 4, is:

Let $A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}$. Then the sum of all elements of the matrix $\operatorname{adj}\left(\operatorname{adj}\left(2(\operatorname{adj} A)^{-1}\right)\right)$ is equal to:

If the system of equations:

$$x + y + z = 5$$

$$x + 2y + 3z = 9$$

$$x + 3y + \lambda z = \mu$$

has infinitely many solutions, then the value of $\lambda + \mu$ is:

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $\det(A - \alpha I) = 0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x - p)^2 + (y - 2p)^2 = 320$, intersects the co-ordinate axes at

Let $A$ be a $3 \times 3$ matrix such that

$$A^T \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}, \; A^T \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \; A \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ 4 \end{bmatrix} \text{ and } A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}.$$

If $\det(A) = 1$, then $\det(\operatorname{adj}(A^2 + A))$ is equal to:

Consider the system of linear equations in $x, y, z$:

$$x + 2y + tz = 0,$$ $$6x + y + 5tz = 0,$$ $$3x + t^2 y + f(t)z = 0,$$

where $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t \in \mathbb{R}$, then $f$

If $f : \mathbf{N} \to \mathbf{Z}$ is defined by

$$f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}, \; k \in \mathbf{N},$$

and $\displaystyle\sum_{n=1}^{k} f(n) = 98$, then $k$ is equal to :

Let $M$ be a $3 \times 3$ matrix such that

$$M\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \; M\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \text{ and } M\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}. \text{ If } M\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 7 \\ 11 \end{bmatrix},$$

then $x + y + z$ equals :

Let $A=\begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ satisfy

$$A^2+\alpha\left(adj\left(adj\left(A\right)\right)\right)+\beta\left(adj\left(A\right)\left(adj\left(adj\left(A\right)\right)\right)\right)=\begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}$$

for some $\alpha, \beta \in \mathbb{R}$.

Then $(\alpha-\beta)^2$ is equal to ________

The sum of all possible values of $\theta \in [0, 2\pi]$, for which the system of equations : $$x\cos 3\theta - 8y - 12z = 0$$ $$x\cos 2\theta + 3y + 3z = 0$$ $$x + y + 3z = 0$$ has a non-trivial solution, is equal to :

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $B = [b_{ij}],\ 1 \le i, j \le 3$. If $B = A^{99} - I$, then the value of $\frac{b_{31} - b_{21}}{b_{32}}$ is :

If the system of linear equations :

$$x + y + z = 6,$$ $$x + 2y + 5z = 10,$$ $$2x + 3y + \lambda z = \mu$$

has infinitely many solutions, then the value of $\lambda + \mu$ equals :

Let $A = \begin{bmatrix} \alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \operatorname{adj}(A)$. If $\det(B) = 66$, then $\det(\operatorname{adj}(A))$ equals :

Let $\mathrm{a} \in \mathbf{R}$ and A be a matrix of order $3 \times 3$ such that $\det(\mathrm{A}) = -4$ and $\mathrm{A} + \mathrm{I} = \begin{bmatrix} 1 & \mathrm{a} & 1 \\ 2 & 1 & 0 \\ \mathrm{a} & 1 & 2 \end{bmatrix}$, where I is the identity matrix of order $3 \times 3$. If $\det((\mathrm{a}+1)\operatorname{adj}((\mathrm{a}-1)\mathrm{A}))$ is $2^\mathrm{m} 3^\mathrm{n}$, $\mathrm{m}, \mathrm{n} \in \{0, 1, 2, \ldots, 20\}$, then $\mathrm{m} + \mathrm{n}$ is equal to :

Let $\mathrm{A} = \begin{bmatrix} \alpha & -1 \\ 6 & \beta \end{bmatrix}$, $\alpha > 0$, such that $\det(\mathrm{A}) = 0$ and $\alpha + \beta = 1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(\mathrm{I} + \mathrm{A})^8$ is :

If the system of linear equations

$$3x + y + \beta z = 3$$ $$2x + \alpha y - z = -3$$ $$x + 2y + z = 4$$

has infinitely many solutions, then the value of $22\beta - 9\alpha$ is :

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha$, $\beta$, and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to :

If the system of equations $$2x+\lambda y+3z=5$$ $$3x+2y-z=7$$ $$4x+5y+\mu z=9$$ has infinitely many solutions, then $(\lambda^2+\mu^2)$ is equal to :

Let $A$ be a matrix of order $3 \times 3$ and $|A| = 5$. If $\left|2\,adj\left(3A\,adj\left(2A\right)\right)\right| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha + \beta + \gamma$ is equal to

If $y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix}$, $x \in \mathbb{R}$, then $\frac{d^2y}{dx^2} + y$ is equal to

Let I be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A} = \begin{bmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{bmatrix}$, $|\mathrm{A}| = -1$. Let $B$ be the inverse of the matrix $adj\left(A\, adj\left(A^2\right)\right)$. Then $|(\lambda \mathrm{B} + \mathrm{I})|$ is equal to ____________

Let $\mathrm{A} = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$, $\mathrm{A}^2 = \mathrm{A}^{\mathrm{T}}$, then the sum of the diagonal elements of the matrix $(\mathrm{A} + \mathrm{I})^3 + (\mathrm{A} - \mathrm{I})^3 - 6\mathrm{A}$ is equal to ________.

Let the matrix $A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$ satisfy $A^n=A^{n-2}+A^2-I$ for $n\geq 3$. Then the sum of all the elements of $A^{50}$ is :

Let the system of equations :

$2x + 3y + 5z = 9$,

$7x + 3y - 2z = 8$,

$12x + 3y - (4 + \lambda)z = 16 - \mu$,

have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4x = 3y$ is

Let $A$ be a $3 \times 3$ matrix such that $|\,adj\,(adj\,(adj\,\mathrm{A}))| = 81$. If

$$S = \left\{ n \in \mathbb{Z} : \left( |adj\,(adj\,A)| \right)^{\frac{(n-1)^2}{2}} = |A|^{\left(3n^2 - 5n - 4\right)} \right\},$$

then $\displaystyle\sum_{n \in S} \left| A^{\left(n^2 + n\right)} \right|$ is equal to

The number of singular matrices of order 2, whose elements are from the set $\{2, 3, 6, 9\}$, is __________.

Let the system of equations $$x + 5y - z = 1$$ $$4x + 3y - 3z = 7$$ $$24x + y + \lambda z = \mu$$ $\lambda, \mu \in \mathbf{R}$, have infinitely many solutions. Then the number of the solutions of this system, if $x, y, z$ are integers and satisfy $7 \leq x + y + z \leq 77$, is :

Let $\mathrm{A} = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix}$. If $\det(adj\,(adj\,(3A))) = 2^m \cdot 3^n$, $m, n \in \mathbb{N}$, then $m + n$ is equal to

Let A be a square matrix of order 3 such that $\det(A)=-2$ and $\det(3\,\mathrm{adj}(-6\,\mathrm{adj}(3A)))=2^{m+n} \cdot 3^{mn}$, $m>n$. Then $4m+2n$ is equal to ________.

If the system of equations :

$x + y + 2z = 6$,

$2x + 3y + az = a + 1$,

$-x - 3y + bz = 2b$,

where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to :

For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \dfrac{1}{2}$ and trace $(A) = 3$. If $B = \mathrm{adj}(\mathrm{adj}(2A))$, then the value of $|B| + \text{trace}(B)$ equals :

If the system of equations

$(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5$

$\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7$

$(\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9$

has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to

If $A$, $B$, and $(\mathrm{adj}(A^{-1}) + \mathrm{adj}(B^{-1}))$ are non-singular matrices of same order, then the inverse of $A(\mathrm{adj}(A^{-1}) + \mathrm{adj}(B^{-1}))^{-1}B$, is equal to

The system of equations $$x + y + z = 6,$$ $$x + 2y + 5z = 9,$$ $$x + 5y + \lambda z = \mu,$$ has no solution if

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$, $A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ and $A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, then $a_{23}$ equals :

If the system of equations

$2x - y + z = 4$

$5x + \lambda y + 3z = 12$

$100x - 47y + \mu z = 212$,

has infinitely many solutions, then $\mu - 2\lambda$ is equal to

Let A be a $3 \times 3$ matrix such that $X^T A X = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}$, $A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}$, and $\det(\operatorname{adj}(2(A + I))) = 2^{\alpha}\, 3^{\beta}\, 5^{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is _____.

For some a, b, let $f(x) = \begin{vmatrix} a + \dfrac{\sin x}{x} & 1 & b \\ a & 1 + \dfrac{\sin x}{x} & b \\ a & 1 & b + \dfrac{\sin x}{x} \end{vmatrix}$, $x \neq 0$, $\lim\limits_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to :

If the system of equations

$x + 2y - 3z = 2$

$2x + \lambda y + 5z = 5$

$14x + 3y + \mu z = 33$

has infinitely many solutions, then $\lambda + \mu$ is equal to :

Let M denote the set of all real matrices of order $3 \times 3$ and let $S = \{-3, -2, -1, 1, 2\}$. Let $S_1 = \{A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall\, i, j\}$, $S_2 = \{A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall\, i, j\}$, $S_3 = \{A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall\, i, j\}$. If $n(S_1 \cup S_2 \cup S_3) = 125\alpha$, then $\alpha$ equals ____.

Let $\mathrm{A} = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix}$ and $\mathrm{P} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, $\theta > 0$. If $\mathrm{B} = \mathrm{PAP^T}$, $\mathrm{C} = \mathrm{P^T B^{10} P}$ and the sum of the diagonal elements of $\mathrm{C}$ is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is :

If the system of equations

$$x+(\sqrt{2}\sin\alpha)y+(\sqrt{2}\cos\alpha)z=0$$ $$x+(\cos\alpha)y+(\sin\alpha)z=0$$ $$x+(\sin\alpha)y-(\cos\alpha)z=0$$

has a non-trivial solution, then $\alpha \in \left(0, \frac{\pi}{2}\right)$ is equal to :

Let $\alpha \in (0, \infty)$ and $A=\begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. If $\det(\mathrm{adj}(2A-A^{T})\cdot\mathrm{adj}(A-2A^{T}))=2^{8}$, then $(\det(A))^{2}$ is equal to :

Let A be a $3 \times 3$ matrix of non-negative real elements such that $A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=3\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$. Then the maximum value of $\det(A)$ is ________.

Let $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = I + \operatorname{adj}(A) + (\operatorname{adj} A)^2 + \ldots + (\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

Let $A$ be a $2 \times 2$ symmetric matrix such that $A \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix}$ and the determinant of $A$ be 1. If $A^{-1} = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals _________

Let A and B be two square matrices of order 3 such that $|A| = 3$ and $|B| = 2$. Then $\left|A^T A (\operatorname{adj}(2A))^{-1} (\operatorname{adj}(4B)) (\operatorname{adj}(AB))^{-1} A A^T\right|$ is equal to :

If the system of equations $$11x + y + \lambda z = -5$$ $$2x + 3y + 5z = 3$$ $$8x - 19y - 39z = \mu$$ has infinitely many solutions, then $\lambda^4 - \mu$ is equal to :

The values of $m$, $n$, for which the system of equations

$x + y + z = 4$,

$2x + 5y + 5z = 17$,

$x + 2y + mz = n$

has infinitely many solutions, satisfy the equation :

Let $\alpha\beta \neq 0$ and $A = \begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix}$. If $B = \begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}$ is the matrix of cofactors of the elements of $A$, then $\det(AB)$ is equal to :

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\ 3r - 2 & 3 & \frac{n(3n-1)}{2} \end{vmatrix}$. Then $2A_{10} - A_8$ is

Let $\alpha\beta\gamma = 45$; $\alpha, \beta, \gamma \in \mathbb{R}$. If $x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$ for some $x, y, z \in \mathbb{R}$, $xyz \neq 0$, then $6\alpha + 4\beta + \gamma$ is equal to ________

If $A$ is a square matrix of order 3 such that $\det(A)=3$ and $\det(\operatorname{adj}(-4\ \operatorname{adj}(-3\ \operatorname{adj}(3\ \operatorname{adj}((2A)^{-1})))))=2^{\mathrm{m}}\, 3^{\mathrm{n}}$, then $\mathrm{m}+2\mathrm{n}$ is equal to :

If the system of equations $$2x+7y+\lambda z=3$$ $$3x+2y+5z=4$$ $$x+\mu y+32z=-1$$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to __________ :

Let $A=\begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}$. If $A^3=4A^2-A-21I$, where $I$ is the identity matrix of order $3\times 3$, then $2a+3b$ is equal to

Let $A=\begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to __________.

If the system of equations $x + 4y - z = \lambda$, $7x + 9y + \mu z = -3$, $5x + y + 2z = -1$ has infinitely many solutions, then $(2\mu + 3\lambda)$ is equal to :