JEE Matrices and Determinants PYQs with Solutions PDF, Check

For the matrices $$A=\begin{bmatrix}3  -4 \\1  -1 \end {bmatrix}$$ and $$B=\begin{bmatrix}-29  49 \\-13  18 \end{bmatrix}$$, if  $$\left(A^{15} + B \right) \begin{bmatrix}x \\y\end{bmatrix} = \begin{bmatrix}0 \\0 \end{bmatrix}$$, then among the following which one is true ?

For some $$\alpha,\beta\epsilon R$$, let $$A=\begin{bmatrix}\alpha &  2 \\ 1 &  2 \end{bmatrix}\text{ and }B=\begin{bmatrix}1 &  1 \\1 &   \beta \end{bmatrix}$$ be such that $$A^{2}-4A+2I=B^2-3B+I=0$$. Then $$(det(adj(A^3-B^3)))^2$$ is equal to _______.

If $$X=\begin{bmatrix}x \\y \\z \end{bmatrix}$$ is a solution of the system of equations AX= B, where adj $$A= \begin{bmatrix}4 & 2 & 2 \\-5 & 0 & 5 \\1 & -2 & 3 \end{bmatrix}$$ and $$B=\begin{bmatrix}4 \\0 \\2 \end{bmatrix}$$, then |x+y+z| is equal to :

Let |A|=6, Where A is a $$3\times3$$ matrix. If $$|adj(3adj(A^{2}\cdot adj(2A)))|=2^{m}\cdot3^{n},m,n\epsilon N$$, then m+n is equal to:

If $$A=\begin{bmatrix}2 & 3 \\3 & 5 \end{bmatrix}$$, then the determinant of the matrix $$ (A^{2025}-3A^{2024}+ A^{2023})$$ is

Let A be a $$3 \times 3$$ matrix such that A+ A^{T} = 0. If $$A\begin{bmatrix} 1 \\-1 \\ 0 \end{bmatrix}=\begin{bmatrix} 3 \\3 \\ 2 \end{bmatrix},A^{2}\begin{bmatrix} 1 \\-1 \\ 0 \end{bmatrix}=\begin{bmatrix} -3 \\19 \\ -24 \end{bmatrix}$$ and $$det(adj(2 adj(A+I))) = (2)^{\alpha }\cdot (3)^{\beta}\cdot (11)^{\gamma},\alpha,\beta,\gamma$$ are non-negative integers, then $$\alpha+\beta+\gamma$$ is equal to _____

The number of $$3\times 2$$ matrices A, which can be formed using the elements of the set {-2, -1 , 0, 1, 2} such that the sum of all the diagonal elements of $$A^{T}A$$ is 5, is_____

Let $$f(x)=\int_{}^{} \frac{7x^{10}+9x^{8}}{(1+x^{2}+2x^{9})^{2}}dx, x>0, \lim_{x \rightarrow 0}f(x)=0$$ and $$f(1)=\frac{1}{4.}$$ If $$A= \begin{bmatrix}0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^{2} & 4 & 1 \end{bmatrix}$$ and B = adj(adj A) be such that |B| = 81 , then $$\alpha^{2}$$ is equal to

Let $$P[P_{ij}]$$ and $$Q=[q_{ij}]$$ be two square matrices of order 3 such that $$q_{ij}= 2^{(i+j-1)}p_{ij}$$ and $$\det (Q)=2^{10}.$$ Then the value of det(adj(adj P)) is:

Let A, Band C be three $$2\times 2$$ matrices with real entries such that $$B=(I+A)^{-1}$$ and A+C=1. If $$BC=\begin{bmatrix}1 & -5 \\-1 & 2 \end{bmatrix}$$ and $$CB\begin{bmatrix}x_{1}\\ x_{2} \end{bmatrix}=\begin{bmatrix}12\\-6 \end{bmatrix}$$, then $$x_{1}+x_{2}$$ is

Let $$A = \begin{bmatrix}0 & 2 & -3 \\-2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix}$$ and B be a matrix such that $$B(I- A)=I+A.$$ Then the sumof the diagonal elements of $$B^{T}B$$ is equal to _________

Let $$A=\begin{bmatrix}3 & -4 \\1 & -1 \end{bmatrix}$$ and B be two matrices such that $$A^{100}=100B+I$$. Then the sum of all the elements of $$B^{100}$$ is_______

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

Let $$ A $$ be a square matrix of order 3 such that $$det(A)=-2 \text{ and }det(3adj(-6adj(3A)))=2^{m+n}\cdot3^{mn}$$, $$m>n. \text{ Then } 4m+2n\text{ is equal to } $$_______

Let $$A = [a_{ij}]$$ be $$3\times 3$$ 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 :

$$ \text{Let } A \text{ be a } 3\times 3 \text{ matrix such that } X^TAX=0 \text{ for all nonzero } 3\times1 \text{ matrices } X=\begin{bmatrix}x\\y\\z\end{bmatrix}. \text{ If } A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix}, \; A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix}, \text{ and } \det(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma, \; \alpha,\beta,\gamma\in\mathbb{N}, \text{ then } \alpha^2+\beta^2+\gamma^2 \text{ is:} \underline{\hspace{2cm}}$$

If the system of equations $$\begin{aligned}x + 2y - 3z &= 2, \\2x + \lambda y + 5z &= 5, \\14x + 3y + \mu z &= 33\end{aligned}$$ has infinitely many solutions, then $$\lambda + \mu \text{ is equal to:} $$

Let  $$A=[a_{ij}]$$  be a square matrix of order 2 with entries either 0 or 1. Let $$E$$  be the event that  $$A$$  is an invertible matrix.  Then the probability  $$P(E)$$ is:

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

Let $$A = [a_{ij}] = \begin{bmatrix}\log_{5}{128} & \log_{4}5 \\\log_{5}8 & \log_{4}25 \end{bmatrix}$$. If $$A_{ij}$$ is the cofactor of $$a_{ij},C_{jk} = \sum_{k=1}^{2}a_{ik}A_{ik},1 \leq i,j \leq 2$$,and $$C = [C_{ij}],$$ then $$8|C|$$ is equal to :

Let $$S = \left\{m \in Z : A^{m^{2}}+A^{m} = 3I - A^{-6}\right\}$$, where $$ A =\begin{bmatrix}2 & -1 \\1 & 0 \end{bmatrix}$$. Then n(S) is equal to ______.

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

Let A = $$[a_{ij}]$$ be a matrix of order $$3 \times 3$$, with $$a_{ij}$$ = $$(\sqrt{2})^{i+j}$$. If the sum of all the elements in the third row of $$A^{2}$$ is $$\alpha + \beta\sqrt{2}, \quad \alpha,\beta \in \mathbb{Z}$$, then $$\alpha + \beta$$ is equal to:

Let A be a square matrix of order 2 such that |A| = 2 and the sum of its diagonal elements is −3. If the points (x, y) satisfying $$A^2 + xA + yI = O$$ lie on a hyperbola whose length of semi major axis is x and semi minor axis is y, eccentricity is e and the length of the latus rectum is l, then $$81(e^4 + l^2)$$ is equal to ______.

Let A be a 3×3 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 ______.

If the system of equations
3x + y + 4z = 3
$$2x+\alpha y-z = -3$$
x+ 2y + z = 4
has no solution, then the value of $$\alpha$$ is equal to :

Let n be the number obtained on rolling a fair die. If the probability that the system
x - ny + z = 6
x + (n - 2)y + (n + 1)z = 8
(n - 1)y + z = 1
has a unique solution is $$\frac{k}{6}$$, then the sum of k and all possible values of n is:

Among the statements :
I: If $$ \begin{vmatrix}1 & \cos\alpha & \cos\beta \\\mathbf{\cos\alpha} & 1 & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & 1\end{vmatrix}=\begin{vmatrix}0 & \mathbf{\cos\alpha}&\mathbf{\cos\beta} \\\mathbf{\cos\alpha} & 0 & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & 0\end{vmatrix}$$, then $$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\frac{3}{2}$$, and

II: $$\begin{vmatrix}x^{2}+x & x+1 & x-2 \\2x^{2}+3x-1 & 3x & 3x-3 \\x^{2}+2x+3 & 2x-1 & 2x-1\end{vmatrix} = px + q$$, then $$p^{2}=196q^{2}$$

The system of linear equations
$$x + y + z = 6$$
$$2x + 5y + az =36$$
$$x + 2y + 3z = b$$

If the system of linear equations : $$x+y+2z=6\\2x+3y+az=a+1\\-x-3y+bz=2b$$ where $$a,b \in R$$, has infinitely many solutions, then 7a + 3b is equal to :

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

If the system of equations $$\begin{aligned} 2x - y + z &= 4, \\ 5x + \lambda y + 3z &= 12, \\ 100x - 47y + \mu z &= 212 \end{aligned}$$ has infinitely many solutions, then $$\mu - 2\lambda$$ is equal to:

$$\text{For some } a,b,\text{ let }f(x)=\left|\begin{matrix}a+\dfrac{\sin x}{x} & 1 & b \\a & 1+\dfrac{\sin x}{x} & b \\a & 1 & b+\dfrac{\sin x}{x}\end{matrix}\right|,x\neq 0,\lim_{x\to 0} f(x)=\lambda+\mu a+\nu b,\text{ Then } (\lambda+\mu+\nu)^2 \text{ is equal to:}$$

Let M and m respectively be the maximum and the minimum value of
$$f(x) =\begin{vmatrix}\mathbf{1+\sin^{2}x} & \mathbf{\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{1+\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{\cos^{2}x} & \mathbf{1+4\sin 4x}\end{vmatrix}$$, $$x \in R$$ then $$M^{4}-m^{4}$$ is equal to :

Let $$A =[a_{ij}]$$ be a 2$$\times$$2 matrix such that $$a_{ij} \in \left\{0,1\right\}$$ for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of x is :

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

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:

If $$A = \begin{pmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{pmatrix}$$, $$B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$, $$C = ABA^T$$ and $$X = A^TC^2A$$, then $$\det X$$ is equal to:

The values of $$\alpha$$, for which $$\begin{vmatrix} 1 & \frac{3}{2} & \alpha + \frac{3}{2} \\ 1 & \frac{1}{3} & \alpha + \frac{1}{3} \\ 2\alpha + 3 & 3\alpha + 1 & 0 \end{vmatrix} = 0$$, lie in the interval

Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$$ and $$|2A|^3 = 2^{21}$$ where $$\alpha, \beta \in Z$$, Then a value of $$\alpha$$ is

If $$f(x) = \begin{vmatrix} 2\cos^4 x & 2\sin^4 x & 3 + \sin^2 2x \\ 3 + 2\cos^4 x & 2\sin^4 x & \sin^2 2x \\ 2\cos^4 x & 3 + 2\sin^4 x & \sin^2 2x \end{vmatrix}$$ then $$\frac{1}{5}f'(0)$$ is equal to ________.

Consider the system of linear equation $$x + y + z = 4\mu$$, $$x + 2y + 2\lambda z = 10\mu$$, $$x + 3y + 4\lambda^2 z = \mu^2 + 15$$, where $$\lambda, \mu \in \mathbb{R}$$. Which one of the following statements is NOT correct?

Consider the system of linear equations $$x + y + z = 5$$, $$x + 2y + \lambda^2 z = 9$$ and $$x + 3y + \lambda z = \mu$$, where $$\lambda, \mu \in R$$. Then, which of the following statement is NOT correct?

If $$f(x) = \begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$$ for all $$x \in \mathbb{R}$$, then $$2f(0) + f'(0)$$ is equal to

Let $$A$$ be a $$3 \times 3$$ matrix and $$\det(A) = 2$$. If $$n = \det(\underbrace{adj(adj(\ldots adj(A)))}_{\text{2024 times}})$$, then the remainder when $$n$$ is divided by 9 is equal to

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 $$\sum_{r=1}^{n} A_r$$ 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 $$\det(\text{adj}(-4 \text{adj}(-3 \text{adj}(3 \text{adj}((2A)^{-1}))))) = 2^m 3^n$$, then $$m + 2n$$ 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 ________