Determinants Questions - JEE

If det($A$) = 0, then det($A^2$) is:

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ 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

The value of the determinant of a matrix is given by the expression. If the matrix is singular, what is the value of the determinant?

If A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}, then A^2 - 4A - 5I is equal to

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

If determinant $\Delta$ = 0, then rank of matrix is:

If $2s = a + b + c$ and the determinant $\begin{vmatrix} a^2 & (s-a)^2 & (s-a)^2 \\ (s-b)^2 & b^2 & (s-b)^2 \\ (s-c)^2 & (s-c)^2 & c^2 \end{vmatrix} = ks^3(s-a)(s-b)(s-c)$, then the numerical quantity $k$ should be

If A = \begin{pmatrix} 4 & 1 \\ 7 & 2 \end{pmatrix} and B = \begin{pmatrix} 2 & -1 \\ 7 & 4 \end{pmatrix}, then B^T A^T is

If $\begin{bmatrix} 1/25 & 0 \\ x & 1/25 \end{bmatrix} = \left( \begin{bmatrix} 5 & 0 \\ -a & 5 \end{bmatrix}^{-1} \right)^2$, then the value of x is

If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0 where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then :

The number of $3\times2$ matrices $A$, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^TA$ is 5, is

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices: $T_p = \left\{ A = \begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \{0, 1, 2, \dots, p-1\} \right\}$. The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\det(A)$ divisible by $p$ is -

Let $M$ and $N$ be square matrices of the same order satisfying $MN = M$ and $NM = N$. Then $(M^{2024} + N^{2024})^{2025}$ is equal to

The number of values of $k$ for which the system of equations$(k+1)x + 8y = 4k$$kx + (k+3)y = 3k - 1$has no solution, is

Let A + 2B = ⎡⎣1 2 0⎤⎦ and 2A - B = ⎡⎣2 -1 5⎤⎦ and 6 -3 3 ⎤⎦ and 2 -1 6 ⎤⎦. If tr(A) denotes the sum of all diagonal elements of the matrix A, then tr(A) - tr(B) has value equal to(JEE Main 2021)

If the trivial solution is the only solution of the system of equations $x - ky + z = 0$, $kx + 3y - kz = 0$, $3x + y - z = 0$, then the set of all values of $k$ is :

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 $\frac{x}{x-m} + \frac{y}{y-n} + \frac{z}{z-r}$ is

If $\begin{vmatrix} 3^2+k & 4^2 & 3^2+3+k \\ 4^2+k & 5^2 & 4^2+4+k \\ 5^2+k & 6^2 & 5^2+5+k \end{vmatrix} = 0$, then the value of $k$ is

Solve the system $x+y+z=6$, $x-y+z=2$, $2x+y-z=1$ using matrix method.

If $\begin{vmatrix} x+3 & 1 & -2 \\ 3 & -2 & 1 \\ -x & -3 & 3 \end{vmatrix} = 0$, find $x$.

If X, Y and Z are positive numbers such that Y and Z have respectively 1 and 0 at their unit's place and \begin{vmatrix} X & 4 & 1 \\ Y & 0 & 1 \\ Z & 1 & 0 \end{vmatrix} is divisible by 10, then X has at its unit's place:

The digits A, B, C are such that the three digit numbers A88, 6B8, 86C are divisible by 72. The determinant \begin{vmatrix} A & 6 & 8 \\ 8 & B & 6 \\ 8 & 8 & C \end{vmatrix} is divisible by

If in the determinant $\begin{vmatrix} x & 3 & 3 \\ 3 & 3 & x \\ 2 & 3 & 3 \end{vmatrix}$, $C_{11} = C_{22}$, where $C_{ij}$ is cofactor of element $a_{ij}$ then $x =$

Find the nature of solution for the given system of equations:$x + 2y + 3z = 1; 2x + 3y + 4z = 3; 3x + 4y + 5z = 0$

If $p + q + r = 0 = a + b + c$, then the value of the determinant $\begin{vmatrix} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{vmatrix}$ is

Let $A = \begin{bmatrix} 1 & 2 & -3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$ and $\text{adj } A = \begin{bmatrix} 1 & -2 & 7 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{bmatrix}$. Find the element $A_{13}$ of $A^{-1}$.

If $a, b, c$ are non-zero real numbers and if the system of equations:$(a-1)x = y + z$$(b-1)y = z + x$$(c-1)z = x + y$has a non-trivial solution, then $ab + bc + ca$ equals

Let $A=\begin{pmatrix}1&0&0\\\sqrt{a}&1&0\\a\sqrt{a}&\sqrt{b}&1\end{pmatrix}$; $a,b\in\mathbb{R}^+$. If for some $n\in\mathbb{N}$, $A^n=\begin{pmatrix}1&0&0\\72&1&0\\3600&72&1\end{pmatrix}$, then the number of triangles formed by joining the vertices of an $n$-sided polygon having no side common with the polygon is

If the system of linear equations x + y + z = 6, x + 2y + 3z = 10, 3x + 2y + \lambda z = m has more than two solutions, then m - \lambda^2 is equal to __________.

The system of equations \[\alpha x - y - z = \alpha - 1\] \[x - \alpha y - z = \alpha - 1\] \[x - y - \alpha z = \alpha - 1\] has no solution if $\alpha$ is

If $a, b, c$ and $d$ are the roots of the equation $x^4 - 2x^3 - 4x^2 - 8x + 16 = 0$, the value of the determinant $\begin{vmatrix} 1-a & 1 & 1 & 1 \\ 1 & 1-b & 1 & 1 \\ 1 & 1 & 1-c & 1 \\ 1 & 1 & 1 & 1-d \end{vmatrix}$ is

Find the value of determinant \[\begin{vmatrix} 13-3\sqrt{5} & 25 & 5 \\ 15-26 & 5 & 10 \\ 3-65 & 15 & 5 \end{vmatrix}\]

If $a \geq b \geq c$ and the system of equations $ax + by + cz = 0$, $bx + cy + az = 0$, $cx + ay + bz = 0$ has a non-trivial solution, then what can be said about the roots of the quadratic equation $at^2 + bt + c = 0$?

The relation between m, n and p is

If determinant is unchanged after row operation, then operation is:

If w ≠ 1 is the complex cube root of unity and matrix H = ⎡⎣1 0⎤⎦0 w, then H70 is equal to

If \[\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy\] where $i = \sqrt{-1}$, then

If $A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}$ and B is the adjoint of A, find the value of $|AB + 2I|$, where I is the identity matrix of order 3.

If a, b, c, d, e, and f are in G.P., then the value of $\begin{vmatrix} a^2 & d^2 & x \\ b^2 & e^2 & y \\ c^2 & f^2 & z \end{vmatrix}$ depends on

Let $A = \{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\}$, where \[P = \begin{pmatrix}1 & 2 & 1\\-2 & 3 & -4\\1 & 9 & -1\end{pmatrix}\]Then the set A

The system of linear equations $x+y+z=6$ $2x+5y+az=36$ $x+2y+3z=b$ has

The number of values of $k$ for which the linear equations$4x + ky + 2z = 0$$kx + 4y + z = 0$$2x + 2y + z = 0$possess a non-zero solution is

If A is a square matrix such that A2 = A, then det(A) is equal to

If the system of linear equations$2x + 2y + 3z = a$$3x - y + 5z = b$$x - 3y + 2z = c$where $a, b, c$ are non-zero real numbers, has more than one solution, then:

If $u + 2v + 3w = 6$, $4u + 5v + 6w = 12$, and $6u + 9v = 4$, then $u + v + w$ is equal to

If $1 + \sin 2x$ $\cos 2x$ $4\sin^2 x$ $\sin^2 x$ $1 + \cos^2 x$ $4\sin^4 x$ $\sin x$ $\cos x$ $1 + 4\sin^2 x$ is evaluated, the maximum value is

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum value of $\dfrac{\det(A)}{b}$ is:

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix} \] and x = a + 2p, y = b + 2q, z = c + 2r, then $\Delta$ equals:

The total number of distinct $x \in \mathbb{R}$ for which\[\begin{vmatrix} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{vmatrix} = 10\]is ______.

Determinants Questions (2072)