Matrices & Determinants Questions - JEE

$A = [a_{ij}]_{n \times n}$ be a square matrix, $n$ is odd such that $a_{ij} = (-1)^j C_j^n C_i^n$, then trace $(A) = $ (Trace $(A)$ denotes sum of diagonal elements of $A$)

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

System of equations $ax + 4y + z = 0, 2y + 3z = 1, 3x - bz = -2$ then which of the following is not true

For $a, b, c, x, y, z \in \mathbb{R}$, if $\Delta_1 = \begin{vmatrix} (a-x)^2 & (b-x)^2 & (c-x)^2 \\ (a-y)^2 & (b-y)^2 & (c-y)^2 \\ (a-z)^2 & (b-z)^2 & (c-z)^2 \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} (1+ax)^2 & (1+bx)^2 & (1+cx)^2 \\ (1+ay)^2 & (1+by)^2 & (1+cy)^2 \\ (1+az)^2 & (1+bz)^2 & (1+cz)^2 \end{vmatrix}$ then $|\Delta_1/\Delta_2| = $

The number of real values of $\lambda$, for which the system of linear equations:$2x + 4y - \lambda z = 0$$4x + \lambda y + 2z = 0$$\lambda x + 2y + 2z = 0$has infinitely many solutions, is

Let A be a square matrix of order 3 such that $\text{adj. }(\text{adj. }(\text{adj. }A)) = \begin{bmatrix}16 & 0 & -24\\ 0 & 4 & 0\\ 0 & 12 & 4\end{bmatrix}$. Find $\text{adj. }A$.

For any real values of $X, Y, Z, L, M, N$ value of $\begin{vmatrix} \cos(X - L) & \cos(X - M) & \cos(X - N) \\ \cos(Y - L) & \cos(Y - M) & \cos(Y - N) \\ \cos(Z - L) & \cos(Z - M) & \cos(Z - N) \end{vmatrix} =$

If $c_{22}c_{33} - c_{23}c_{32} = \det(A^{20}) = 2^{20} \equiv 2^m$, find the value of $m$.

If $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $P = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$, $Q = P^T AP$, then $PQ^{2014}P^T =$

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 true?

Eigen values of matrix $\begin{bmatrix} 2 & 1 & 1 \\ 2 & 3 & 4 \\ -1 & -1 & -2 \end{bmatrix}$ are:

If $A$ is a square matrix of order $n$, then $(\underbrace{\text{adj adj}\cdots\text{adj}A}_{(n-1)\text{ times}})\cdot(\underbrace{\text{adj adj}\cdots\text{adj}A}_{n\text{ times}})$ is equal to

For Problems 14 and 15$A$ and $B$ are square matrices such that det.$(A) = 1$, $BB^T = I$, det.$(B) > 0$, and $A(\text{adj.}A + \text{adj.}B) = B$.$AB^{-1} =$

If $A=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$, then $A^{-1}=$

$A=\begin{bmatrix}1&-2\\0&1\end{bmatrix}$, $B=\begin{bmatrix}5&2\\2&1\end{bmatrix}$. Then $(ABA^T)^5(AB^TA^T)^{10}=X$. Then trace of matrix $X$ is

The values of $k \in R$ for which the system of equations $x + ky + 3z = 0$, $kx + 2y + 2z = 0$, $2x + 3y + 4z = 0$ has nontrivial solution are

If $a_1, a_2, a_3, \ldots, a_n, \ldots$ are in GP, then $\Delta = \begin{vmatrix} \log a_n & \log(a_n r) & \log(a_n r^2) \\ \log(a_n r^3) & \log(a_n r^4) & \log(a_n r^5) \\ \log(a_n r^6) & \log(a_n r^7) & \log(a_n r^8) \end{vmatrix}$ equals:

The value of the determinant $\begin{vmatrix} ka & k^2+a^2 & 1 \\ kb & k^2+b^2 & 1 \\ kc & k^2+c^2 & 1 \end{vmatrix}$ is

Let for $i = 1, 2, 3$, $p_i(x)$ be a polynomial of degree 2 in $x$, $p_i'(x)$ and $p_i''(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let,\[A(x) = \begin{bmatrix} p_1(x) & p_1'(x) & p_1''(x) \\ p_2(x) & p_2'(x) & p_2''(x) \\ p_3(x) & p_3'(x) & p_3''(x) \end{bmatrix}\]and $B(x) = [A(x)]^T A(x)$. Then determinant of $B(x)$

If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}$, then $AB$ is equal to:

If determinant of identity matrix is 1, its rank is:

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

If $\begin{vmatrix} x^n & x^{n+2} & x^{n+3} \\ y^n & y^{n+2} & y^{n+3} \\ z^n & z^{n+2} & z^{n+3} \end{vmatrix} = (x-y)(y-z)(z-x)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)$, then $n$ equals

Let \[f(x) = \begin{vmatrix} \cos(x+x^2) & \sin(x+x^2) & -\cos(x+x^2) \\ \sin(x-x^2) & \cos(x-x^2) & \sin(x-x^2) \\ \sin 2x & 0 & \sin(2x^2) \end{vmatrix}\]Find the value of $f'(0)$.

If det($A$) = 2, then det(Adj $A$) for 2 \times 2 matrix is:

Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB - BA$ is:

If $\det(A)$=4, then det(A\)^3A⁻^2) equals:

If $A(\alpha, \beta) = \begin{bmatrix} \cos\alpha & \sin\alpha & 0 \\ -\sin\alpha & \cos\alpha & 0 \\ 0 & 0 & e^\beta \end{bmatrix}$, then $A(\alpha, \beta)^{-1}$ is equal to

Let $\omega$ be complex cube root of unity. Let $S = \begin{bmatrix} 1 & a & b \\ \omega^1 & 1 & c \\ \omega^2 & \omega^1 & 1 \end{bmatrix}$, where each of $a, b, c$ are either $\omega$ or $\omega^2$. Then number of distinct non singular possible such matrices $S$ is _____.

If $A = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ 7 & 2 \end{bmatrix}$, find $3A - 2B$.

$A = \begin{bmatrix} \frac{1}{2}[x] & |\sin y| \\ \cos z & 1 \end{bmatrix}$, $B = \begin{bmatrix} [x] & [y] \\ [z] & 1 \end{bmatrix}$ if $x \in [-2, 2]$, $y, z \in (-\pi, \pi)$ if number of triplets $(x, y, z)$ such that $A = B$ is $k$, then value of $k/7$ is _____.

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

Let $A$ be a $3 \times 3$ matrix which contains five 'a' & four 'b' then number of symmetric matrices possible is $k$, number of zeros at the end of $k!$ is _____.

If \det(A)=4, then $\det(A$\)^{-1}$A$^T) equals:

Let $P = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

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

Let A and B 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 with B is commutative.

If the system of linear equations$x + ky + 3z = 0$$3x + ky - 2z = 0$$2x + 4y - 3z = 0$has a non-zero solution $(x, y, z)$, then $\dfrac{xz}{y^2}$ is equal to

Number of positive integral solutions of the equation $\begin{vmatrix} x^3 + 1 & x^2y & x^2z \\ xy^2 & y^3 + 1 & y^2z \\ x^2 & yz^2 & z^3 + 1 \end{vmatrix} = 30$ are

Let $A = \begin{bmatrix} 1 & \frac{-1-i\sqrt{3}}{2} \\ \frac{-1+i\sqrt{3}}{2} & 1 \end{bmatrix}$. Then $A^{(0)} = 2^k \cdot A$ where $k$ is

If the system of linear equations $x + y + z = 0$, $3x + ky - 2z = 0$, $2x + 4y - 3z = 0$ has a non-zero solution $(x, y, z)$, then $\frac{xz}{y^2}$ is equal to

If $S_r = a^r + b^r + c^r$ the value of $\begin{vmatrix} S_0 & S_1 & S_2 \\ S_1 & S_2 & S_3 \\ S_2 & S_3 & S_4 \end{vmatrix}$ is equal to $(\alpha - \beta)^2(\beta - \gamma)^{2k-2}(\gamma - a)^{k^2-2}$. Then $k$ is

Consider three matrices $A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix}$ and $C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}$, then the value of the sum $tr(A) + tr\left(\frac{ABC}{2}\right) + tr\left(\frac{A(BC)^2}{4}\right) + tr\left(\frac{A(BC)^3}{8}\right) + \ldots \infty$ is

If A is order 3 square matrix such that $|A| = 2$, then $|\text{adj}(\text{adj}(\text{adj } A))|$ is

If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A+Bx)(x-A)^2$ then $A + 2B$ equals

If $A = \begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}$ and $B$ is the adjoint of $A$, then $\det(AB+2I)$ is (where $I$ is $3\times3$ identity)

Let $k$ be a positive real number and let $A = \begin{bmatrix} 2k & 2\sqrt{k} & 2\sqrt{k} \\ 2\sqrt{k} & 1 & -2k \\ -2\sqrt{k} & 2k & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k-1 & \sqrt{k} \\ 1-2k & 0 & 2\sqrt{k} \\ -\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix}$. If $\det(\text{adj }A) + \det(\text{adj }B) = 10^6$, then greatest integer of $k$ is equal to

Let $M$ be a $3 \times 3$ matrix satisfying $$M \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}, M \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$$ and $$M \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix}$$. Then the sum of diagonal entries of $M$ is ____

Let $\omega$ be the complex number $\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}$. Then, the number of distinct complex numbers $z$ satisfying $$\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix} = 0$$ is equal to ____

The system $(1-\lambda)x+3y-4z=0$, $x-(3+\lambda)y+5z=0$, $3x+y-\lambda z=0$ possesses non-trivial solutions for

Matrices & Determinants Questions (2045)