Determinants Questions - JEE

Let A = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} and B = \begin{pmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & -14^2 \\ -15^2 & 16^2 & 17^2 \end{pmatrix}, then the value of A'BA is:

If x ≠ y ≠ z & x, y, z are in GP and D = 0, then y is equal to -

Let $p, q, r$ be real numbers such that $p + q + r \neq 0$. The system of linear equations$x + 2y - 3z = p$$2x + 6y - 11z = q$$x - 2y + 7z = r$has at least one solution if :

Matrix A = x321y422z, if xyz = 60 and 8x + 4y + 3z = 20, then A(adj A) is equal to -

The number of real values of x satisfying <mfenced open="|

$\text{tr}(A)$ is equal to

Let A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}, d = |A| \neq 0, |A - d(\text{adj } A)| = 0. Then

If P = \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix}, then P^{50} is:

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)

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

22. $D = \begin{vmatrix} 10^4 + 2 & 10^7 + 3 & 10^8 + 8 \\ 10^9 + 9 & 10^2 + 8 & 10^3 - 4 \\ 10^3 - 5 & 10^8 + b & 10^6 + a \end{vmatrix}$ where $a, b$, both $\in \{1,2,3,4,5,6,7,8,9\}$Number of ordered pairs $(a, b)$ such that $D = 2n + 1, n \in Z$ is

If $\det(A)$=4, then det(A\)^5(A\)⁻^5 + $I)(A$⁻^5 - \(I)) equals:

If $\det(A)$=4, then det(A\)^5(A\)⁻^5 + $I)(A$⁻^5 - $I)(A$⁻^5 + \(I)) equals:

If x ≠ y ≠ z & x, y, z are in A.P. and D = 0, then 2xy^2z + x^2z^2 is equal to-

If sin22xcos2x4sin2x2tan2x2cos2x-sin2x-2cos4xtan2x2sin4x = a0 + a1(cosx) + a2(cos2x) + ........ + an(cosn x), then a0 is -

If $\det(A)$=4, then det(A\)^5(I\) - $A$⁻^5)(I\) + $A$⁻^5)(I\) - $A$⁻^5)) equals:

The determinant cos(θ+ϕ)-sin(θ+ϕ)cos2ϕsinθcosθsinϕ-cosθsinθcosϕ is -

If $\det(A)$=5, then det(A\)^6(I\) - $A$⁻^6)(I\) + $A$⁻^6)(I\) - $A$⁻^6)) equals:

For $\alpha,\beta\in\mathbb{R}$, suppose the system $x-y+z=5$, $2x+2y+\alpha z=8$, $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are roots of:

If $\det(A)$=k, then det(I\)\cdotA) equals:

If $A$ is identity matrix, then det(A\)^n) equals:

If $\det(A)$=k, then det(A\)^0) equals:

Let $A = \begin{pmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{pmatrix}$. If $AA^T = I_3$, then $|p|$ is

If a2 + b2 + c2 = -2 and f(x) = 1+a2x(1+b2)x(1+c2)x(1+a2)x1+b2x(1+c2)x(1+a2)x(1+b2)x1+c2x then f(x) is a polynomial of degree-

Let \(0

If determinant is product of two matrices, then:

If $\det(A)$=5, then det(A\)^5A^3A⁻^7) equals:

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If\[\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2-1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k,\]then k is equal to

If det($A$) = 3, then det(Adj $A$) for 3 \times 3 matrix is:

94. Let A and B are square matrices of same order satisfying $AB = A$ and $BA = B$, then $(A^{2019} + B^{2019})^{2020}$ is equal to:

996. Let $P$ be a $2 \times 2$ matrix such that $P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}$ and $P^2\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}$. If $x_1$ and $x_2$ are two values of $x$ for which $|P - xI| = 0$, where $I$ is an identity matrix of order 2, then find the value of $x_1^2 + x_2^2$.

Sum of elements of (adjA) B is -

Given: $x + ay + z = 3,\; x + 2y + 2z = 6,\; x + 5y + 3z = b$.For no solution, which of the following is correct?

The set of equations $\lambda x - y + (\cos\theta)z = 0$, $3x + y + 2z = 0$, $(\cos\theta)x + y + 2z = 0$, \(0 \leq \theta

Let $X = (ABA^T)^{2020}$ where $B$ is symmetric. If $X^T = ((ABA^T)^{mm})^2 - ((ABA^T)^{2020}) - (ABA^T)^{2020}$ and $X^T = X$ is a symmetric matrix, then $a_{21} = b_{13} = c_1$, $b_0 = c_2$. Required value = ?

If \det(A)=k, then \det(c$A$^T) for $n \times n$ matrix equals:

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

If det($A$) = -4, then det($A^T$) is:

Let \[\Delta = \begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = ax - 12\]Then the value of $a$ is:

If det($A$) = -3, then det($A^3$) is:

Given $a = x/(y-z)$, $b = y/(z-x)$, and $c = z/(x-y)$, where $x, y$ and $z$ are not all zero, then the value of $ab + bc + ca$ is

Question 87: Statement-1: The value of the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix} = 2$.Statement-2: Neither of two rows or columns of [a matrix property regarding determinants].

The system of equations x + y + z = 5; x + 2y + 3z = 9; x + 3y + Dz = I is called lazy, if it has no solution. The condition for this is

If $A$ is a $3 \times 3$ matrix with \det(A)=a, then \det(adj(adj \(A)) equals:

Evaluate the determinant $\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ac & ab \end{vmatrix}$.

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

Let $A = [a_{ij}]_{n \times n}$ where $a_{ij} = i^2 - j^2$. Then $A$ is

Let a, b, and c be such that b(a + c) ≠ 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

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

Consider a matrix A(θ) = sinθcosθ-cosθsinθ then

Determinants Questions (2072)