Determinants Questions - JEE

If $A = \begin{bmatrix} 1 & 3 \\ 3 & 2 \\ 2 & 5 \end{bmatrix}$ & $B = \begin{bmatrix} -1 & -2 \\ 0 & 5 \\ 3 & 1 \end{bmatrix}$ and $A + B - D = O$ (zero matrix), then $D$ matrix will be-

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

The number of values of $\theta\in[0,\pi]$ for which the system $3x-y+3z=2$, $x+y+4z=1$, $-6x+y+\lambda z=-3$ has unique solution is

The system of equations $x_1 - x_2 + x_3 = 2, 3x_1 - x_2 + 2x_3 = -6$ and $3x_1 + x_2 + x_3 = -18$ has

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

Let $\begin{vmatrix}a&\sqrt{5}&\sqrt{7}\\\sqrt{3}&b&\sqrt{7}\\\sqrt{3}&\sqrt{5}&c\end{vmatrix}=0$, ($a\neq\sqrt{3}, b\neq\sqrt{5}, c\neq\sqrt{7}$) and $\dfrac{a}{a-\sqrt{3}}+\dfrac{b}{b-\sqrt{5}}+\dfrac{c}{c-\sqrt{7}}=\lambda$. If $a=2\sqrt{3}$, then the point $(b^2,c^2)$ may lie on the line

If $\Delta = \begin{vmatrix} 2a & 3a + 2x^2 & 4a + 3x^2 + 2x^4 \\ 2a & 3a + 2x^2 & 4a + 3x^2 + 2x^4 \\ 3a & 6a + 3x^2 & 10a + 6x^2 + 3x^4 \end{vmatrix}$

Which of the following matrices is NOT left inverse of matrix $\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ 2 & 3 \end{bmatrix}$?

Consider, $A = \begin{bmatrix} a & 2 & 1 \\ 0 & b & 0 \\ 0 & -3 & c \end{bmatrix}$, where $a$, $b$ and $c$ are the roots of the equation $x^3 - 3x^2 + 2x - 1 = 0$. If matrix $B$ is such that $AB = BA$, $A + B - 2I \neq O$ and $A^2 - B^2 = 4I - 4B$, then find the value of $\det(B)$.

If θ ∈ R, then maximum value of $\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+\sin\theta & 1 \\ 1 & 1 & 1+\cos\theta \end{vmatrix}$ is

If the system of linear equations x - 4y + 7z = g 3y - 5z = h -2x + 5y - 9z = k is consistent, then :

If $A$ and $B$ are square matrices of size $n \times n$ such that $A^2 - B^2 = (A - B)(A + B)$, then which of the following will be always true?

Let $ A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix} $ and $ B = A^{20} $. Then the sum of the elements of the first column of $ B $ is

Let $P = \begin{pmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{pmatrix}$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

Number of 3 × 3 symmetric matrices which can be formed by three '0', three '1' & three '-1' only, is

If A = $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ & A$^n$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, (where n $\ge$ 2 & n $\in$ N), then -

For Problems 4–6If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then$(A + I)^5$ is equal to (where $I$ is identity matrix)

Let A be a square matrix such that $A(\text{adj. }A) = \begin{bmatrix}4 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 4\end{bmatrix}$. Find the values of:(i) |adj. A|

If $A$ and $B$ are symmetric matrices of the same order and $X = AB + BA$ and $Y = AB - BA$, then $(XY)^T$ is equal to

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has exactly two distinct solutions is

If $\alpha, \beta, \gamma$ are the angles of a triangle and the system of equations \[\cos(\alpha-\beta)x + \cos(\beta-\gamma)y + \cos(\gamma-\alpha)z = 0\] \[\cos(\alpha+\beta)x + \cos(\beta+\gamma)y + \cos(\gamma+\alpha)z = 0\] \[\sin(\alpha+\beta)x + \sin(\beta+\gamma)y + \sin(\gamma+\alpha)z = 0\] has non-trivial solutions, then triangle is necessarily

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all nonsingular matrices of the form $\begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \theta & 1 \end{bmatrix}$, where each of $a, b,$ and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of a 3 × 3 matrix A and |A| = 4, then $\alpha$ is equal to:

First row of a matrix A is [1 3 2]. If adj. $A = \begin{bmatrix} -2 & 4 & \alpha \\ -1 & 2 & 1 \\ 3\alpha & -5 & -2 \end{bmatrix}$, then det.(A) is

Let A and B be two $2 \times 2$ matrices. Consider the statements:(i) $AB = O \Rightarrow A = O$ or $B = O$(ii) $AB = I_2 \Rightarrow A = B^{-1}$(iii) $(A+B)^2 = A^2 + 2AB + B^2$Then

If A and B are two non-singular matrices of order 3 such that $AA^T = 2I$ and $A^{-1} = A^T - A \cdot \text{adj}(2B^{-1})$, then $\det(B)$ is equal to

If $A_1, B_1, C_1, \ldots$ are, respectively, the cofactors of the elements $a_1, b_1, c_1, \ldots$ of the determinant $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$, $\Delta \ne 0$, then the value of $\begin{vmatrix} B_2 & C_2 \\ B_3 & C_3 \end{vmatrix}$ is equal to

If a2 + b2 + c2 = -2 and f(x) = <mfenced open="|

If a, b, c are in AP, then x+1x+2x+ax+2x+3x+bx+3x+4x+c equals -

Matrices $A$ and $B$ satisfy $AB = B^{-1}$, where $B = \begin{bmatrix}2 & -1\\ 2 & 0\end{bmatrix}$. Without finding $B^{-1}$, find the value of $K$ for which $KA - 2B^{-1} + I = O$.

If $A$ and $B$ are two invertible matrices of the same order, then $\text{adj}(AB)$ is equal to

The determinant a2a2-(b-c)2bcb2b2-(c-a)2cac2c2-(a-b)2ab is divisible by -

If $A = \begin{bmatrix} a+ib & c+id \\ -c+id & a-ib \end{bmatrix}$ and $a^2 + b^2 + c^2 + d^2 = 1$, then $A^{-1}$ is equal to

If $\Delta = \begin{vmatrix} 3 & 4 & 5 & x \\ 4 & 5 & 6 & y \\ 5 & 6 & 7 & z \\ x & y & z & 0 \end{vmatrix} = 0$, then

If $A$ and $B$ are square matrices such that $A^{2006} = O$ and $AB = A + B$, then $\det(B)$ equals

For the system of linear equation2x - y + 3z = 53x + 2y - z = 74x + 5y + αz = βWhich of the following is(are) CORRECT?

If det(A)=5, then det(($A$^T)^{-1}) equals:

For Problems 1–3Let $A$ be a matrix of order $2 \times 2$ such that $A^2 = O$.$A^2 - (a+d)A + (ad - bc)I$ is equal to

Let the system of equations$x + 5y - z = 1$$4x + 3y - 3z = 7$$24x + y$+$\lambda z$= $\mu$$\lambda $, $\mu$$\ in $R , have infinitely many solutions. Then the number of the solutions of this system, If x, y, z are integers and satisfy 7$\le$$x + y + z$$\le$77, is

If $A_1 = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}$, $A_2 = \begin{bmatrix} 0 & 0 & 0 & i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{bmatrix}$, then $A_i A_k + A_k A_i$ is equal to

Find the number of real roots of the equation \[\begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix} = 0,\] where $a \neq b \neq c$ and $b(a+c) > ac$.

If $A$, $B$ and $C$ are angles of a triangle, then the value of $\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}$ is

If $n = 1$ and the system of equations $x + y - 1 = 0$, $2x - y - c = 0$, and $bx + 3by - c = 0$ is consistent, then the possible real values of $b$ are

If $\begin{bmatrix} \cos\dfrac{2\pi}{7} & -\sin\dfrac{2\pi}{7} \\ \sin\dfrac{2\pi}{7} & \cos\dfrac{2\pi}{7} \end{bmatrix}^k = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then the least positive integral value of $k$ is

Let A be a 2 × 2 matrix with non-zero entries and let $A^2 = I$, where I is a 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.Statement 1: Tr(A) = 0Statement 2: |A| = 1

If <mfenced open="|

If $\begin{bmatrix}\alpha & \beta\\ \gamma & -\alpha\end{bmatrix}$ is to be the square root of two-rowed unit matrix, then $\alpha,\ \beta,$ and $\gamma$ should satisfy the relation

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

Find non-zero values of $x$ satisfying the matrix equation:\[x\begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2\begin{bmatrix} 8 & 5x \\ 4 & 4x \end{bmatrix} = 2\begin{bmatrix} x^2+8 & 24 \\ 10 & 6x \end{bmatrix}\]

If the system of equations x + y + z = 6 2x + 5y + αz = β x + 2y + 3z = 14 has infinitely many solutions, then α + β is equal to :

Determinants Questions (2072)