Determinants Questions (2072)

If $A$, $B$, $C$ are the angles of a triangle and $\begin{vmatrix} 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A + \sin^2 A & \sin B + \sin^2 B & \sin C + \sin^2 C \\ 1+\sin A & 1+\sin B & 1+\sin C \end{vmatrix} = 0$, then triangle $ABC$ is

A and B are two matrices such that the order of A is $3 \times 4$, if $A'B$ and $BA'$ are both defined, then

$(I + A)^{100} =$

If $\alpha$ and $\beta$ are the roots of the equation $|\begin{matrix} 1 & 5 \\ -4 & 7 \end{matrix}| \begin{matrix} \frac{x}{19} & -\frac{3}{19} \\ \frac{1}{19} & \frac{5}{19} \end{matrix} = \begin{matrix} 1 & 3 \\ -4 & 7 \end{matrix} \begin{matrix} \frac{x^2 - 5x + 6}{-3} \end{matrix} = -4|$, then the value of $(2 - \alpha)(2 - \beta)$ is

If $M = \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix}$ and $N = \begin{bmatrix} 0 & 5 \\ 2 & 0 \end{bmatrix}$, then $M^{2011}$ is -

If $A = \begin{bmatrix} -2 & -1 & 1 \\ -1 & 7 & 4 \\ 1 & -x & -3 \end{bmatrix}$ be symmetric matrix then find the value of $x$.

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 greatest value of $\frac{xyz}{(x-m)(y-n)(z-r)}$ is

$A = [a_{ij}]_{m \times n}$ is a square matrix, if

If the coordinates of the vertices of an equilateral triangle with sides of length a are (x₁, y₁), (x₂, y₂) and (x₃, y₃), then $$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$$ is equal to

If $[1 \quad x \quad 2] \begin{bmatrix} 2 & 3 & 1 \\ 0 & 4 & 2 \\ 0 & 3 & 2 \end{bmatrix} \begin{bmatrix} x \\ 1 \\ -1 \end{bmatrix} = O$, then the value of $x$ is

The system of equations has a non-trivial solution if and only if $\begin{vmatrix} \sin 3\theta & -2 & 3 \\ \cos 2\theta & 8 & -7 \\ 2 & 14 & 11 \end{vmatrix} = 0$

If $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$, then $A^5 =$

If $\alpha, \beta, \gamma$ are real numbers, then prove that $$D = \begin{vmatrix} 1 & \cos(\beta-\alpha) & \cos(\gamma-\alpha) \\ \cos(\alpha-\beta) & 1 & \cos(\gamma-\beta) \\ \cos(\alpha-\gamma) & \cos(\beta-\gamma) & 1 \end{vmatrix} = 0$$

If $[1 \quad x \quad 2] \begin{bmatrix} 2 & 3 & 1 \\ 0 & 4 & 2 \\ 0 & 3 & 2 \end{bmatrix} \begin{bmatrix} x \\ 1 \\ -1 \end{bmatrix} = O$, then the value of $x$ is

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

If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then $(A + B)^9$ is equal to

The number of real values of x satisfying x3x+22x-12x-14x3x+17x-217x+612x-1 = 0 is -

If 1, \omega and \omega^2 are the cube roots of unity, then \begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix} is equal to

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

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 :

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

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

$(I + A)^{100} =$

If \(\det(A)\)=2, then det(A\)⁻^2A^5A⁻^3) equals:

A square matrix $P$ satisfies $P^2 = I - P$, where $I$ is the identity matrix. If $P^n = 5I - 8P (n \in N)$, then minimum value of $n$ is equal to

3. If px^4 + qx^3 + rx^2 + sx + t = x2+3xx-1x+3x+12-xx-3x-3x+43x then t is equal to -

Let $A = \begin{bmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{bmatrix}$. If $AA^T = I^3$, then $|p|$ is :

If two rows of a determinant are interchanged, then determinant becomes:

If \(\det(A)\)=3, then det(A\)^5A^2A⁻^2A⁻^5A^3A⁻^3) equals:

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (adj A)^{20} is equal to

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

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

If \(\det(A)\)=k, then det((A\)^0)^{-1}) equals:

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

If A = 5!6!7!6!7!8!7!8!9!, then |adj(adj(2A))| is equal to :

Let $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. If $B = \begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$, then the sum of all the elements of the matrix $\sum_{n=1}^{50} B^n$ is equal to

If \(\det(A)\)=2, then det(A\)^3A^3A⁻^5) equals:

Find the sum of all positive integral values of a for which every solution to the system of equation x + ay = 3 and ax + 4y = 6 satisfy the inequalities x > 1, y > 0.

If determinant is expanded along any row, result is:

Let a, b, c are the solutions of the cubic x3 - 5x2 + 3x - 1 = 0, then find the value of the determinant

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

If x + 3 2y + x z - 1 4w - 8 = -x - 1 0 3 2w, then find the value of \(x + y + z + w\).

If \(\det(A)\)=k, then det(A\)^n(A\)⁻^n + \(I)^2) equals:

Matrices A and B satisfy \(AB = B^{-1}\), where \(B = \begin{pmatrix} 2 & -2 \\ -1 & 0 \end{pmatrix}\). Find the value of \(\lambda\) for which \(\lambda A - 2B + I = O\), without finding \(B^{-1}\).

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

The total number of matrices A = 02y12xy-12x-y1, (x, y ∈ R, x ≠ y) for which AT A = 3I3 is :-

If \(\det(A)\)=k, then det(A\)^nA^{-m}) where n=m equals:

If \(\det(A)\)=2, then det(A\)^3(A\)⁻^3 + \(I)(A\)⁻^3 - \(I)) equals:

If \(\det(A)\)=k, then det(A\)^nA^mA⁻^nA⁻^mA^nA⁻^nA^mA⁻^m) equals:

If $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$, show that $A^k = \begin{bmatrix} 1+2k & -4k \\ k & 1-2k \end{bmatrix}$, where $k$ is any positive integer.