Determinants Questions - JEE

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

Let $A=I_2-MM^T$, where $M$ is a real matrix of order $2\times1$ such that the relation $M^TM=I_1$ holds. If $\lambda$ is a real number such that the relation $AX=\lambda X$ holds for some non-zero real matrix $X$ of order $2\times1$, then the sum of squares of all possible values of $\lambda$ 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

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

The set of all values of $\lambda$ for which the system of linear equations$x - 2y - 2z = \lambda x$$x + 2y + z = \lambda y$$-x - y = \lambda z$has a non-trivial solution:

If $\Delta_1 = \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} x & \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x & 1 \\ \cos 2\theta & 1 & x \end{vmatrix}$, $x \neq 0$; then for all $\theta \in \left(0, \dfrac{\pi}{2}\right)$:

If $f(x) = \begin{vmatrix} \sec^2 x & 1 & 1 \\ \cos^2 x & \cos^2 x & \csc^2 x \\ 1 & \cos^2 x & \cot^2 x \end{vmatrix}$, then which statement is correct?

$A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix}$ then $B^T A^T$ is

Let $P = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & -1 \\ 3 & 1 & 1 \end{bmatrix}$. If the product $PQ$ has inverse $R = \begin{bmatrix} -1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2 \end{bmatrix}$, then $Q^{-1}$ equals

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + x + 1 = 0$. Then for $y \neq 0$ in $R$,\[\begin{vmatrix} y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{vmatrix}\] is equal to:

If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then which of the following is true?

Let $a, b, c$ be such that $b(a+c) \neq 0$. If \[\begin{vmatrix} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{vmatrix} + \begin{vmatrix} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2}a & (-1)^{n+1}b & (-1)^n c \end{vmatrix} = 0,\] then the value of $n$ is

If $\Delta = \begin{vmatrix} a^2 & b\sin A & C\sin A \\ b\sin A & 1 & \cos A \\ C\sin A & \cos A & 1 \end{vmatrix}$ is independent of which variable? (where $a, b, c$ are sides of a triangle and $A, B, C$ are opposite angles)

If $A = \begin{bmatrix} -4 & -4 \\ 3 & 1 \end{bmatrix}$, then the determinant of the matrix $(A^{2016} - 2A^{2015} - A^{2014})$ is

If $P$, $Q$ and $R$ represent the angles of an acute angled triangle, then the value of \[ A = \begin{vmatrix} 1 & 1+\sin P & \sin P(1+\sin P) \\ 1 & 1+\sin Q & \sin Q(1+\sin Q) \\ 1 & 1+\sin R & \sin R(1+\sin R) \end{vmatrix} \text{ is} \]

If $\alpha, \beta \neq 0$, and $f(n) = \alpha^n + \beta^n$ and\[\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = k(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2,\]then k is equal to

If one of the roots of the equation \begin{vmatrix} 7 & 6 & x^2-13 \\ 2 & x^2-13 & 2 \\ x^2-13 & 3 & 7 \end{vmatrix} = 0 is $x = 2$, then the sum of all other five roots is:

Let $\{D_1, D_2, D_3, \ldots, D_n\}$ be the set of third-order determinants that can be made with the distinct non-zero real numbers $a_1, a_2, \ldots, a_9$. Then

If A is a square matrix of order 5 and $2A^{-1} = A^T$, then the remainder when $|\text{adj}(\text{adj}(\text{adj}\, A))|$ is divided by 7 is

The number of 3 x 3 matrices A whose entries are either 0 or 1 and for which the system A[x y z]^T = [1 0 0]^T has exactly two distinct solutions is:

If the system of linear equations has a non-zero solution:\[\begin{align} x + 2ay + az &= 0 \\ x + 3by + bz &= 0 \\ x + 4cy + cz &= 0 \end{align}\] then $a, b, c$:

For Problems 19–21Given that the system of equations $x = cy + bz$, $y = az + cx$, $z = bx + ay$ has nonzero solutions and at least one of the $a, b, c$ is a proper fraction.System has solution such that

If A and B are two non-zero $n \times n$ matrices such that $A^2 + B = A^2 B$, then

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:

Which of the following values of $x$ satisfy the equation$\begin{vmatrix} (1-x)^2 & (1-2x)^2 & (1-3x)^2 \\ (2-x)^2 & (2-2x)^2 & (2-3x)^2 \\ (3-x)^2 & (3-2x)^2 & (3-3x)^2 \end{vmatrix} = -648x$?

Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\det(ABA^T) = 8$ and $\det(AB^{-1}) = 8$, then $\det(BA^{-1}B^T)$ is equal to:

Let $\alpha = \dfrac{\pi}{5}$ and $A = \begin{bmatrix} \cos\dfrac{\pi}{5} & \sin\dfrac{\pi}{5} \\ -\sin\dfrac{\pi}{5} & \cos\dfrac{\pi}{5} \end{bmatrix}$, then find $\det(A + A^2 + A^3 + A^4)$

Find the value of $x, y, z$ and $w$ which satisfy the matrix equation $\begin{bmatrix} x+3 & 2y+x \\ z-1 & 4w-8 \end{bmatrix} = \begin{bmatrix} -x-1 & 0 \\ 3 & 2w \end{bmatrix}$.

Let A and B be two invertible matrices of order 3 × 3. If det(ABAT) = 8 and det(AB−1) = 8, then det(BA−1BT) is equal to

We have $\displaystyle\sum_{r=1}^{n-1} \Delta_r = \Delta_1 + \Delta_2 + \cdots + \Delta_{n-1}$. Evaluate the sum of determinants and find its value.

If $a, b$ and $c$ are unequal, what is the condition that the value of the determinant, $\Delta = \begin{vmatrix} a & a & a+1 \\ b & b & b+1 \\ c & c & c+1 \end{vmatrix}$ is $0$?

Let $\lambda$ and $\alpha$ be real. Then the number of integral values of $\lambda$ for which the system of linear equations$\lambda x + (\sin\alpha)y + (\cos\alpha)z = 0$$x + (\cos\alpha)y + (\sin\alpha)z = 0$$-x + (\sin\alpha)y - (\cos\alpha)z = 0$has non-trivial solutions is

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

If \[\begin{vmatrix} a & a^2 & 1 + a^3 \\ b & b^2 & 1 + b^3 \\ c & c^2 & 1 + c^3 \end{vmatrix} = 0\] and vectors $(1, a, a^2)$, $(1, b, b^2)$ and $(1, c, c^2)$ are non-coplanar, then the product $abc$ equals:

17. Let $M_n = (a_{ij})$ where $i, j = 1, 2, 3, \ldots, n$. We first find out $a_{11}$ for the $n^{\text{th}}$ matrix, which is the $n^{\text{th}}$ term in the series: $1, 2, 6, 15, \ldots$. The diagonal elements of the $n^{\text{th}}$ matrix form an arithmetic progression with first term $1 + \dfrac{n(n-1)(2n-1)}{6}$ and common difference $n+1$. Find the required sum $M_n$.

The equations $(\lambda - 1)x + (3\lambda + 1)y + 2\lambda z = 0$, $(\lambda - 1)x + (4\lambda - 2)y + (\lambda + 3)z = 0$ and $2x + (3\lambda + 1)y + 3(\lambda - 1)z = 0$ give non-trivial solution for some values of $\lambda$, then the ratio $x : y : z$, when $\lambda$ has smallest of these values is:

If $\Delta = \begin{vmatrix} 1 & 3\cos\phi & 1 \\ \sin\phi & 1 & 3\cos\phi \\ 1 & \sin\phi & 1 \end{vmatrix}$, the maximum value of $\Delta$ is

We have \[\Delta_1 = \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}\] and \[\Delta_2 = \begin{vmatrix} x & \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x & 1 \\ \cos 2\theta & 1 & x \end{vmatrix}\] Then $\Delta_1 + \Delta_2$ equals:

Let $x, y, z > 1$ and $A = \begin{pmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 2 & \log_y z \\ \log_z x & \log_z y & 3 \end{pmatrix}$. Then $|\text{adj}(\text{adj}A^2)|$ is equal to

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If a₁b₁c₁, a₂b₂c₂ and a₃b₃c₃ are three digit even natural numbers and $\Delta = \begin{vmatrix} c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3 \end{vmatrix}$, then $\Delta$ is

If P is an orthogonal matrix and $Q = PAP^T$ and $x = P^T Q^{1000} P$, then $x^{-1}$ is, where A is involutary matrix

If $f(x) = a + bx + cx^2$ and $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3 = 1$, then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is equal to

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

If the system of equations $x+2ay+az=0$, $x+3by+bz=0$, $x+4cy+cz=0$ has a non-zero solution, then $a,b,c$

Let $A=[a_{ij}]_{2\times2}$, where $a_{ij}\neq0$ for all $i,j$ and $A^2=I$. Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$. Then $3a^2+4b^2$ is equal to

If $A + B = \text{BA}$ and $A^2 - B^2 = I$, then the value of the determinant of matrix $A^T + B$ (where $A$ and $B$ are square matrices of order $3 \times 3$)

If $a, b, c, \lambda \in \mathbb{N}$, then the least possible value of $\begin{vmatrix} a^2 + \lambda & ab & ac \\ ba & b^2 + \lambda & bc \\ ca & cb & c^2 + \lambda \end{vmatrix}$ is

Let $f(a,b) = \begin{vmatrix} a & a^2 & 0 \\ 1 & 2a + b & (a+b)^2 \\ 0 & 1 & 2a + 3b \end{vmatrix}$. Which is a factor of $f(a,b)$?

For $x \neq y \neq z$, $\begin{vmatrix} 1+x^3 & x^2 & 1 \\ 1+y^3 & y^2 & 1 \\ 1+z^3 & z^2 & 1 \end{vmatrix} = 0$ if $xyz$ is

Determinants Questions (2072)