Determinants Questions - JEE

If $x > m, y > n, z > r$ ($x, y, z > 0$) such that $\begin{vmatrix} x & n & r \\ m & y & r \\ m & n & z \end{vmatrix} = 0$, then the value of $\frac{x}{x-m} - 1 + \frac{y}{y-n} - 1 + \frac{z}{z-r} - 1$ is

If $A^2 = A$, then $(A + I)^5$ is equal to (where $I$ is identity matrix)

If det($A$) = 4 and det($B$) = 2, then det($A^2$B) is:

If determinant has symmetric entries, then best approach is:

Let $f(x) = \begin{vmatrix} x \cos x & 2x \sin x & x \tan x \\ 1 & 2x & 1 \end{vmatrix}$, then $\lim_{x \to 0} \frac{f(x)}{x^2} =$

Let $A = [a_{ij}]_{3 \times 3}$, $B = [b_{ij}]_{3 \times 3}$ where $b_{ij} = 3^{i-j}a_{ij}$, $C = [c_{ij}]_{3 \times 3}$, where $c_{ij} = 4^{i-j}b_{ij}$ be any three matrices. If $|A| = 4$, then $|B| + |C| = (|X|$ denotes determinant of matrix $X)$

Let $A$ be any $3 \times 3$ invertible matrix. Then, which one of the following is not always true?

If $\Delta$ = \[ \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} \], then $\Delta$ equals:

If all elements of a row are multiplied by k, determinant becomes:

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and a = d = g, then $\Delta$ equals:

If determinant has a row with two identical elements, best approach is:

If determinant has two columns proportional, then value is:

If \begin{vmatrix} (x+1) & (x+1)^2 & (x+1)^3 \\ (x+2) & (x+2)^2 & (x+2)^3 \\ (x+3) & (x+3)^2 & (x+3)^3 \end{vmatrix} is expressed as a polynomial in $x$, then the term independent of $x$ is:

If the matrix $\begin{pmatrix} 0.3 & b & c \\ l & m & n \\ 0 & p & q \end{pmatrix}$ is an orthogonal matrix, find the sum of all possible values of $10(mq - np)$.

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

If determinant has a common factor in a row, then determinant becomes:

The value of $(3 \; 2 \; 0) U^{-1} \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$ is

If determinant is such that two rows differ only by a constant factor, then determinant is:

For the matrix $A = \begin{bmatrix} 4 & -4 & 5 \\ -2 & 3 & -3 \\ 3 & -3 & 4 \end{bmatrix}$ find $A^{-2}$.

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 $A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}$ where $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Find $AB$.

Given A2 − A + I = O, then A−1 equals:

If the system of equations $x = 2a$, $y = 3b$, $z = c$ has non-zero solutions, i.e., $\Delta = 0$ where $\Delta = \begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{vmatrix} = 0$, then $a$, $b$, $c$ are in:

If $\begin{vmatrix}\sin x+1&\sin 2x&\sin 3x\\ \sin 2x&\sin 3x+a&\sin 4x\\ \sin 3x&\sin 4x&\sin 5x+a^2\end{vmatrix}=2025(f(x)+45)$ where $f(x)$ is a function of $x$ and $a$ is complex, then sum of all possible values of $a$ is

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?

If $a_1, a_2, a_3, \ldots, a_n, \ldots$ are in GP, then the value of the determinant \[\begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}\] is

If $D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}$, for $x \neq 0,\, y \neq 0$ then $D$ is

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A\,\text{adj}\,A = AA^T$, then $5a + b$ is equal to:

If $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then the matrix $A^{-50}$ when $\theta = \dfrac{\pi}{12}$, 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:

Let $\lambda$ be a real number for which the system of linear equations$x + y + z = 6$$4x + \lambda y - \lambda z = \lambda - 2$$3x + 2y - 4z = -5$has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation

If $x = a$, $y = b$, $z = c$ is a solution of the system of linear equations$x + 8y + 7z = 0$$9x + 2y + 3z = 0$$x + y + z = 0$such that the point $(a, b, c)$ lies on the plane $x + 2y + z = 6$, then $2a + b + c$ equals

Given $(A - 3I)(A - 5I) = 0$ for a matrix $A$, which gives $\alpha A + \beta A^{-1} = 4I$. Then $\alpha + \beta$ equals:

The sum of the elements of $U^{-1}$ is

Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $a, b, c, d \neq 0$. If $A^2 = A$, then the value of $|A|$ (determinant of $A$) is

If $A^2 - A + I = 0$ where $A$ is a square matrix and $I$ is the unit matrix of the same order then $A^{-1}$ is

If x, y and z are the integers in AP lying between 1 and 9 and x51, y41 and z31 are three digits numbers, the value of \begin{vmatrix} x & 51 \\ y & 41 \\ z & 31 \end{vmatrix} \div \begin{vmatrix} x & 5 \\ y & 4 \\ z & 3 \end{vmatrix} is

If $A \times \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} = 6I$ where $I$ is a unit matrix of order $2 \times 2$, then $A$ is

The system of linear equations$x + y + z = 2$$2x + 3y + 2z = 5$$2x + 3y + (a^2 - 1)z = a + 1$

Let $A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$. Which of the following is true?

Let $ A $ be a $ 3 \times 3 $ matrix such that $ A^2 - 5A + 7I = 0 $.Statement-I: $ A^{-1} = \dfrac{1}{7}(5I - A) $.Statement-II: The polynomial $ A^3 - 2A^2 - 3A + I $ can be reduced to $ 5(A - 4I) $.Then:

If $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, then which one of the following statements is not correct?

If $A$ is a $3 \times 3$ matrix such that $|5 \cdot \text{adj}\, A| = 5$, then $|A|$ is equal to

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A - 3I)(A - 5I) = O$, where $I = I_3$ and $O = O_3$. If $\alpha A + \beta A^{-1} = 4I$, then $\alpha + \beta$ is equal to

If $A^k = 0$ (A is nilpotent with index k), $(I - A)^p = I + A + A^2 + \ldots + A^{k-1}$, thus p is,

If $\Delta$ = 7 and two rows are interchanged, new determinant is:

Given $A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}$ where $\theta \in \left(\dfrac{3\pi}{4}, \dfrac{5\pi}{4}\right)$, then $\det(A)$ lies in the interval:

If determinant is non-zero, inverse exists because:

If $\phi(x) = \begin{vmatrix} x^2 + 5x + 3 & 2x + 5 & 3 \\ 3x^2 - x - 4 & 6x - 1 & 9 \\ 7x^2 + 6x + 9 & 14x + 6 & 21 \end{vmatrix} = ax^3 + bx^2 + cx + d$, then

If $a_1b_1c_1$, $a_2b_2c_2$, and $a_3b_3c_3$ 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

Determinants Questions (2072)