Determinants Questions - JEE

Suppose $D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$ and $D' = \begin{vmatrix} a_1+pb_1 & b_1+qc_1 & c_1+ra_1 \\ a_2+pb_2 & b_2+qc_2 & c_2+ra_2 \\ a_3+pb_3 & b_3+qc_3 & c_3+ra_3 \end{vmatrix}$. Then

If $a, b, c$ are different, then the value of $x$ satisfying $\begin{vmatrix} 0 & x^2-a & x^3-b \\ x^2+a & 0 & x^2+c \\ x^4+b & x-c & 0 \end{vmatrix} = 0$ is

The arbitrary constant on which the value of the determinant $\begin{vmatrix} 1 & a & a \\ \cos(\pi - d)a & \cos(\pi a) & \cos(\pi - d)a \\ \sin(\pi - d)a & \sin(\pi a) & \sin(\pi - d)a \end{vmatrix}$ does not depend, is

The number of distinct real values of $K$ such that the system of equations $x+2y+z=1$, $x+3y+4z=K$, $x+5y+10z=K^2$ has infinitely many solutions is

Solve: \[\begin{vmatrix} x^2-1 & x^2+2x+1 & 2x^2+3x+1 \\ 2x^2+x-1 & 2x^2+5x-3 & 4x^2+4x-3 \\ 6x^2-x-2 & 6x^2-7x+2 & 12x^2-5x-2 \end{vmatrix} = 0\]

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

Find the value of the determinant $\begin{vmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & 20 \end{vmatrix}$

If $A$ is matrix of order 3 such that $|A|=5$ and $B=\text{adj}\,A$, then the value of $\left\||A^{-1}|(AB)^T\right\|$ is equal to

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

For Problems 7 and 8Consider an arbitrary $3 \times 3$ non-singular matrix $A = [a_{ij}]$. A matrix $B = [b_{ij}]$ is formed such that $b_{ij}$ is the sum of all the elements except $a_{ij}$ in the $i$th row of $A$.The value of $|B|$ is equal to

For Problems 12 and 13Let for $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$, there be three row matrices $R_1, R_2$ and $R_3$, satisfying the relations, $R_1 A = [1\ 0\ 0]$, $R_2 A = [2\ 3\ 0]$ and $R_3 A = [2\ 3\ 1]$. If $B$ is square matrix of order 3 with rows $R_1, R_2$ and $R_3$ in order, thenThe value of det.(B) is

If $A$ is a square matrix of order 3 such that $\det(A)=3$ and $\det(adj(-4\cdot adj(-3\cdot adj(3\cdot adj((2A)^{-1})))))=2^m3^n$. Then $m+2n$ is equal to

l, m and n are the pth, qth and rth terms of an GP and all positive, then $\begin{vmatrix} \log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1 \end{vmatrix}$ equals

If A is an 3 × 3 non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals:

Let $B^2 = I$ and $AB = \begin{bmatrix} p & x & a \\ b & q & y \\ z & c & r \end{bmatrix}$. Given that $\text{tr}(AB + AB^3 + \cdots + AB^{19}) = 210$, find the number of ordered triplets $(p, q, r)$ where $p, q, r \in \mathbb{N}$ and $p + q + r = 21$.

For Problems 4–6If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then$(A + B)^7$ is equal to

If the system of equations $\alpha x + y + z = \alpha - 1$, $x + \alpha y + z = \alpha - 1$, $x + y + \alpha z = \alpha - 1$ has non-zero solutions, then the condition on $\alpha$ is:

If $\alpha, \beta, \gamma$ are the roots of $px^3 + qx^2 + r = 0$, then the value of the determinant $\begin{vmatrix} \alpha\beta & \beta\gamma & \gamma\alpha \\ \beta\gamma & \gamma\alpha & \alpha\beta \\ \gamma\alpha & \alpha\beta & \beta\gamma \end{vmatrix}$ is

The value of determinant $\begin{vmatrix} \log_a\left(\frac{x}{y}\right) & \log_a\left(\frac{y}{z}\right) & \log_a\left(\frac{z}{x}\right) \\ \log_{a^2}\left(\frac{y}{z}\right) & \log_{a^2}\left(\frac{z}{x}\right) & \log_{a^2}\left(\frac{x}{y}\right) \\ \log_{a^3}\left(\frac{z}{x}\right) & \log_{a^3}\left(\frac{x}{y}\right) & \log_{a^3}\left(\frac{y}{z}\right) \end{vmatrix}$ is

If $A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}$ and $A + A^T = I$, find the value of $\alpha$.

If $A$ is a non-diagonal involutory matrix, then

Let $M$ denote the matrix $\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$, where $i^2 = -1$, and let $I$ denote the identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Then the matrix $I + M + M^2 + M^3 + M^4 + \ldots + M^{2010}$ is equal to:

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?

Matrix $A$ such that $A^2 = 2A - I$, where $I$ is the identity matrix, then for $n \geq 2$, $A^n$ is equal to

\[f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}\] Then find the value of $\displaystyle\lim_{x \to 0} \frac{f(x)}{x}$.

For the system of equations $(k+2)x + 10y = k$ $kx + (k+3)y = k - 1$ to have no solution, the number of values of $ k $ is:

If $A$ is a $3 \times 3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals

If $A, B, C$ are angles of a triangle, then the value of $\begin{vmatrix} e^{2iA} & e^{-iC} & e^{-iB} \\ e^{-iC} & e^{2iB} & e^{-iA} \\ e^{-iB} & e^{-iA} & e^{2iC} \end{vmatrix}$ is

Let $P=\begin{bmatrix}0&2&\lambda\\2&3&1\\1&\mu&3\end{bmatrix}$ and $\text{Adj}(P)=\begin{bmatrix}10&-7&-1\\-5&-1&2\\-5&2&-4\end{bmatrix}$. Then $\left|(\text{adj}P)^{-1}+14\,\text{adj}(P^{-1})\right|$ equals

By using properties of determinants, show that $\begin{vmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{vmatrix} = (5x+4)(4-x)^2$

Let $A = \begin{vmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{vmatrix}$. If $|A^2| = 25$ then $|\alpha|$ equals

If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2$, then the ordered pair $(A, B)$ is equal to

Let $A = \begin{bmatrix} \tan\frac{\pi}{3} & \sec\frac{2\pi}{3} \\ \cot\left(2013\frac{\pi}{2}\right) & \cos(2012\pi) \end{bmatrix}$ and $P$ be a $2 \times 2$ matrix such that $PP^T = I$, where $I$ is an identity matrix of order 2. If $Q = PAP^T$ and $R = [r_{ij}]_{2\times 2} = P^T Q^8 P$, then find $r_{11}$.

Given $A = \begin{pmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{pmatrix}$ and $AA^T = I_3$, find $|p|$.

If matrix $A = \begin{pmatrix} 1 & -1 \\ 1 & -2 \end{pmatrix}$ satisfies $A^n = 5I - 8A$, then $n =$

It is given that each entry of matrix $A$ is an integer. Which of the following is necessarily true?

$A = [a_{ij}]_{n \times n}$ 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 $A^{-1}=$

Determinants Questions (2072)