Matrices & Determinants Questions - JEE

$\begin{vmatrix} ^xC_r & ^xC_{r+1} & ^xC_{r+2} \\ ^yC_r & ^yC_{r+1} & ^yC_{r+2} \\ ^zC_r & ^zC_{r+1} & ^zC_{r+2} \end{vmatrix}$ is equal to

If $\begin{vmatrix} b+c & c+a & a+b \\ a+b & b+c & c+a \\ c+a & a+b & b+c \end{vmatrix} = k \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix}$, then the value of $k$ is

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 $g(x) = (c_1 - x)(c_2 - x)(c_3 - x)$Which of the following is not a constant term in $f(x)$?

Let $A$ and $B$ be two symmetric matrices of order 3.Statement-1: $A(BA)$ and $(AB)A$ are symmetric matrices.Statement-2: $AB$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1. How many of these are symmetric?

Let \(x

If A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}, then \text{adj}(3A + 12A^2) is equal to

The number of solutions of the set of equations \frac{2x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0, \frac{x^2}{a^2} + \frac{2y^2}{b^2} - \frac{z^2}{c^2} = 0, \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{2z^2}{c^2} = 0 is

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19}(A-I) + A^{18}(A-I)^2 + \dots + (A-I)^{20} is equal to

If f(x) = \begin{vmatrix} (1-x)a & (1-2x)b & 1 \\ 1 & (1-x)a & (1-2x)b \\ b & a & (1-2x) \end{vmatrix}, where a, b are positive integers, then:

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (adj A) is equal to

If A = \begin{pmatrix} 2 & 1 \\ -4 & -2 \end{pmatrix}, then I + 2A + 3A^2 + \cdots + \infty is

If A = \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix}, then A is

If A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} and det. (A^n - I) = 1 - \lambda^n, n \in N then the value of \lambda, is -

Let $A_k=[a_{ij}]$ be square matrix of order 3 with $a_{ij}=(i-j)^k$ for all $i,j\in\{1,2,3\}$. Determinant value of $|A_1+A_3+A_5+\cdots+A_{2023}|$ equals

If a, b and c are sides of $\triangle ABC$ such that\[a^3 + b^3\cos B + c^3\cos A + b^2 + c^2 = 0\]\[c^3 + a^3\cos B + a^2\cos A + c^2 + a^2 = 0\]\[b^3 + a^3\cos B + b^3\cos A + a^2 + b^2 = 0\]where $\phi, \psi, \omega \in \mathbb{R}$ and $\angle A, \angle B, \angle C \in [\frac{\pi}{8}, \frac{7\pi}{8}]$, then $\triangle ABC$ is

Let A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}, d = |A| \neq 0, |A - d(\text{adj } A)| = 0. Then

Example 26 (Statement-1): Consider the determinant \[f(x) = \begin{vmatrix} x^2-a & x^3 & 0 \\ x^2+a & 0 & x^2+c \\ x+b & x+c & 0 \end{vmatrix}\] Then $f(x) = 0$ has one root $x = 0$.Statement-2: The value of skew-symmetric determinant of odd order is always zero.

If the value of a third order determinant is 11, find the value of the square of the determinant formed by the cofactors.

The greatest value of c \in \mathbb{R} for which the system of linear equations x - cy - cz = 0, cx - y + cz = 0, cx + cy - z = 0 has a non-trivial solution, is

Let A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} p \\ q \end{pmatrix} \neq \begin{pmatrix} 0 \\ 0 \end{pmatrix} are matrices satisfying AB = B and a + d = 5050. Find the value of (ad - bc).

If A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} and I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, then 10A^{-1} is equal to

Let A = \begin{pmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}, a, b \in \mathbb{R}. If for some n \in \mathbb{N}, A^n = \begin{pmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{pmatrix} then n + a + b is equal to.

Number of values of a for which the system of equations ax + (2 − a)y = 4 + a and ax + (2a − 1)y = a² − 2 possess no solution, is

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (A^T)^{20} is equal to

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

If A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}, then \text{adj}(\text{adj } A) is equal to

Let A be a 3x3 matrix such that A2 = I. If the determinant of A is -1 and the trace of A is 0, then the eigenvalues of A are:

If $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$, show that $A^k = \begin{bmatrix} 1+2k & -4k \\ k & 1-2k \end{bmatrix}$, where $k$ is any positive integer.

If \det(A)=2 and \det(B)=3, then $\det(A$\)^{-1}BA) equals:

If det(A)=3 and det(B)=2, then det($A$^T\)$B$^T) equals:

If $\det(A)$=k, then det(A\)^{-1}$A$^TA) equals:

For each real $x$, \(-1

If $\det(A)$=5, then det(A\)^{-1} + $A$^T) equals:

If $A^T [A|B = A|B]B^T$, then find $|A|$

If $\det(A)$=k, then det(A\)^{-1}$A$^2A^{-1}) equals:

If $\det(A)$=k, then det(A\)^nA⁻^nA^n) equals:

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19}(\text{adj } A) + \dots + A(\text{adj } A)^{19} + (\text{adj } A)^{20} \text{ is equal to}

Let A be a 3x3 matrix such that A^2 = A. If det(A) = 0 and the trace of A is 2, then the rank of A is:

Let f(x) = 1+sin2xcos2x4sin2xsin2x1+cos2x4sin2xsin2xcos2x1+4sin2x, then the maximum value of f(x), is-

If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix}$, $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{x}(A^2 + cA + d)$ then the sum of values of $c$ and $d$ is

If $\det(A)$=4, then det(A\)^6A^3A⁻^3) equals:

If \det(A) = -2, then \det((-\(A)^3) is:

If a determinant has a row of zeros, then its value is:

If $A$ is a square matrix of order $n \times n$ and $k$ is a scalar, then $adj(kA)$ is equal to

If $A$, $B$, $C$ are the angles of a triangle and $\begin{vmatrix} 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A + \sin^2 A & \sin B + \sin^2 B & \sin C + \sin^2 C \\ 1+\sin A & 1+\sin B & 1+\sin C \end{vmatrix} = 0$, then triangle $ABC$ is

A and B are two matrices such that the order of A is $3 \times 4$, if $A'B$ and $BA'$ are both defined, then

$(I + A)^{100} =$

If $\alpha$ and $\beta$ are the roots of the equation $|\begin{matrix} 1 & 5 \\ -4 & 7 \end{matrix}| \begin{matrix} \frac{x}{19} & -\frac{3}{19} \\ \frac{1}{19} & \frac{5}{19} \end{matrix} = \begin{matrix} 1 & 3 \\ -4 & 7 \end{matrix} \begin{matrix} \frac{x^2 - 5x + 6}{-3} \end{matrix} = -4|$, then the value of $(2 - \alpha)(2 - \beta)$ is

If $M = \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix}$ and $N = \begin{bmatrix} 0 & 5 \\ 2 & 0 \end{bmatrix}$, then $M^{2011}$ is -

Matrices & Determinants Questions (2045)