Determinants Questions - JEE

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

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

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

Let $\alpha, \beta$ ($\alpha \neq \beta$) be the values of $m$, for which the equations $x + y + z = 1$; $x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\displaystyle\sum_{n=1}^{10}(n^\alpha + n^\beta)$ 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:

Consider the matrices $A=\begin{pmatrix}2&-5\\3&m\end{pmatrix}$, $B=\begin{pmatrix}20\\m\end{pmatrix}$ and $X=\begin{pmatrix}x\\y\end{pmatrix}$. Let the set of all $m$ for which $AX=B$ has a negative solution (i.e., $x<0$ and $y<0$) be the interval $(a,b)$. Then $8\int_a^b|A|\,dm$ is equal to ________.

Let $\lambda,\mu\in\mathbb{R}$. If the system of equations $3x+5y+\lambda z=3$, $7x+11y-9z=2$, $97x+155y-189z=\mu$ has infinitely many solutions, then $\mu+2\lambda$ is equal to:

Let $A$ be a non-singular matrix of order 3. If $\det(3\,\text{adj}(2\,\text{adj}((\det A)A)))=3^{-13}\cdot2^{-10}$ and $\det(3\,\text{adj}(2A))=2^m\cdot3^n$, then $|3m+2n|$ is equal to

Let $B=\begin{bmatrix}1&3\\1&5\end{bmatrix}$ and $A$ be a $2\times2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta-\alpha$ is equal to:

255. Let $A = [a_{ij}]_{2\times 2}$ be a matrix where $a_{ij} \in \{2, 3\}$. If determinant of matrix $A$ is non-negative, then probability that it is invertible is:

Let $A^T = A$ and $B^T = B$. Consider the following statements:Statement-1: $A(BA)^T = A(BA)$Statement-2: $(AB)^T = B^T A^T = BA$ since $AB$ is commutative.Which of the following is correct?

Prove that $$\begin{vmatrix} 2 & \alpha+\beta+\gamma+\delta & \alpha\beta+\gamma\delta \\ \alpha+\beta+\gamma+\delta & 2(\alpha+\beta)(\gamma+\delta) & \alpha\beta(\gamma+\delta)+\gamma\delta(\alpha+\beta) \\ \alpha\beta+\gamma\delta & \alpha\beta(\gamma+\delta)+\gamma\delta(\alpha+\beta) & 2\alpha\beta\gamma\delta \end{vmatrix} = 0$$

Using factor property of determinants prove that $$\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix} = (x-y)(y-z)(z-x)$$

If a, b, c are sides of a scalene triangle, then the value of determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is always:

61. If the system of equations$x + y + z = 5$$x + 2y + 3z = 9$$x + 3y + \alpha z = \beta$has infinitely many solutions, then $\beta - \alpha$ equals ______.

921. If $A$ and $B$ are square matrices of order 3 such that $2(A + B) = A^T + B^T + 3I$ and $AA^T = 4I$, then find the value of $\det.(12A^{-1} - BA^T + I)$.[Note: $I$ is an identity matrix of order 3 and $P^T$ denotes the transpose of matrix $P$.]

66. If $B = \begin{bmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{bmatrix}$ is the inverse of a $3 \times 3$ matrix $A$, then the sum of all values of $\alpha$ for which $\det(A) + 1 = 0$, is ______.

Let A = [$\alpha$-1 ],$\alpha$> 0 , such that$det(A) = 0$and$\alpha$+$\beta$= 1. If I denotes 2 $\times$ 2 identity matrix, then the 6$\beta$matrix$(1 + A)$is: 8$4 -1$

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

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

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 $A$ be a non-singular idempotent matrix of order $2025\times2025$. Consider statements: (i) Trace of $A$ = 2025, (ii) $A$ has to be a scalar matrix, (iii) Trace of adjoint of $A^2$ = 2025. Which are true?

Given the system of equations$x + y + z = 5$$x + 2y + 3z = 9$$x + 3y + \alpha z = \beta$For infinitely many solutions, find $\beta - \alpha$.

Let $A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$. The only correct statement about the matrix $A$ is

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ where $a, b \in \mathbb{N}$. Then $AB = BA$ implies which of the following?

For Problems 10–12\[f(x) = \begin{vmatrix} x+c_1 & x+a & x+a \\ x+b & x+c_2 & x+a \\ x+b & x+b & x+c_3 \end{vmatrix}\] and $g(x) = (c_1 - x)(c_2 - x)(c_3 - x)$Coefficient of $x$ in $f(x)$ is

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1, 0, 1\}$. Then, the maximum possible value of the determinant of $P$ is ___. (JEE Advanced 2018)

For Problems 22–24Consider the system of equations$x + y + z = 6$$x + 2y + 3z = 10$$x + 2y + \lambda z = \mu$The system has unique solution if

If S is the set of distinct values of 'b' for which the following system of linear equations$x + y + z = 1$$x + ay + z = 1$$ax + by + z = 0$has no solution, then S is

Given the system of linear equations$(1+\alpha)x + \beta y + z = 2$$\alpha x + (1+\beta)y + z = 3$$\alpha x + \beta y + 2z = 2$The number of ordered pairs $(\alpha, \beta)$ for which the system has a unique solution is

If the system of linear equations $x + y + z = 6$, $x + 2y + 3z = 14$, and $2x + 5y + \lambda z = \mu$ ($\lambda, \mu \in \mathbb{R}$) has a unique solution, then

An ordered pair $(\alpha, \beta)$ for which the system of linear equations$(1 + \alpha)x + \beta y + z = 2$$\alpha x + (1 + \beta)y + z = 3$$\alpha x + \beta y + 2z = 2$has a unique solution, is:

Given matrix $B$ is the inverse of matrix $A$, where \[B = \begin{bmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{bmatrix}\] and $\det(A) + 1 = 0$. Find the sum of all values of $\alpha$.

Let S be the set of all real values of k for which the system of linear equations$x + y + z = 2$$2x + y - z = 3$$3x + 2y + kz = 4$has a unique solution. Then S is

If $A\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$, then $A^{-1}$ equals

If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $f(x) = \dfrac{1+x}{1-x}$, then $f(A)$ is

The system of linear equations:$x + \lambda y - z = 0$$\lambda x - y - z = 0$$x + y - \lambda z = 0$has a non-trivial solution for

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x = cy + bz$, $y = az + cx$ and $z = bx + ay$. Then $a^2 + b^2 + c^2 + 2abc$ is equal to

The system of linear equations$x + \mu y - z = 0$$\mu x - y - z = 0$$x + y - \mu z = 0$has a non-trivial solution for:

If the system of linear equations 2x + 2y + 3z = a, 3x - y + 5z = b, x - 3y + 2z = c, where a, b, c are non-zero real numbers, has more than one solution, then

Find the minors and cofactors of elements $-3, 5, -1$ and $7$ in the determinant $\begin{vmatrix} 2 & -3 & 1 \\ 4 & 0 & 5 \\ -1 & 6 & 7 \end{vmatrix}$.

The value of $\sin^{-1}(\det A) + \tan^{-1}(9 \det C)$ is

Let $f(x)=\begin{vmatrix}\cos x&\cos^2x&\cos^4x\\\cos3x&\cos^23x&\cos^43x\\\cos5x&\cos^25x&\cos^45x\end{vmatrix}$, then $\displaystyle\int_0^{\pi}f(x)\,dx$ equals

If $M$ is a square matrix of order 3 such that $|M|=2$, then $\left|\text{adj}\!\left(\dfrac{M}{2}\right)\right|$ equals

If determinant has repeated row operations applied, its value:

If $\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$ where $a, b, c$ are all different, then the determinant $\begin{vmatrix} 1 & 1 & 1 \\ (x-a)^2 & (x-b)^2 & (x-c)^2 \\ (x-b)(x-c) & (x-c)(x-a) & (x-a)(x-b) \end{vmatrix}$ vanishes when

Consider the system of equations:$3x + y - z = 0$ ...(1)$x - \dfrac{py}{4} + z = 0$ ...(2)$2x - y + 2z = q$ ...(3)The number of ordered pairs $(p, q)$ in $[1, 10]$ for which the system has no solution is:

The set of natural number N is partitioned into arrays of rows and columns in the form of matrices as $m_1 = [1], m_2 = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}, m_3 = \begin{bmatrix} 6 & 7 & 8 \\ 9 & 10 & 11 \\ 12 & 13 & 14 \end{bmatrix}, \cdots m_n = \begin{bmatrix} . & . & . \\ . & . & . \\ . & . & . \end{bmatrix}$ and so on, then sum of the elements of the diagonal in $m_{10}$ is ___________.

Let $A = [a_{ij}]_{n \times n}$, $n$ is odd natural number. Then determinant of matrix $(A - A^T)^{2015}$ is _____.

Let $f(n) = \begin{vmatrix} ^n P_n & ^{n+1} P_{n+1} & ^{n+2} P_{n+2} \\ ^n C_n & ^{n+1} C_{n+1} & ^{n+2} C_{n+2} \end{vmatrix}$, where the symbols have their usual meanings. Then $f(n)$ is divisible by

Determinants Questions (2072)