Determinants Questions (2072)

Example 26 (Statement-1): Consider the determinant \[f(x) = \begin{vmatrix} x^2-a & x^3 & 0 \\ x^2+a & 0 & x^2+c \\ x+b & x+c & 0 \end{vmatrix}\] Then \(f(x) = 0\) has one root \(x = 0\).Statement-2: The value of skew-symmetric determinant of odd order is always zero.

If the value of a third order determinant is 11, find the value of the square of the determinant formed by the cofactors.

The greatest value of c \in \mathbb{R} for which the system of linear equations x - cy - cz = 0, cx - y + cz = 0, cx + cy - z = 0 has a non-trivial solution, is

Let A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} p \\ q \end{pmatrix} \neq \begin{pmatrix} 0 \\ 0 \end{pmatrix} are matrices satisfying AB = B and a + d = 5050. Find the value of (ad - bc).

If A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} and I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, then 10A^{-1} is equal to

Let A = \begin{pmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}, a, b \in \mathbb{R}. If for some n \in \mathbb{N}, A^n = \begin{pmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{pmatrix} then n + a + b is equal to.

Number of values of a for which the system of equations ax + (2 − a)y = 4 + a and ax + (2a − 1)y = a² − 2 possess no solution, is

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

If A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} and f(x) = \frac{1+x}{1-x}, then f(A) is

If A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}, then \text{adj}(\text{adj } A) is equal to

Let A be a 3x3 matrix such that A2 = I. If the determinant of A is -1 and the trace of A is 0, then the eigenvalues of A are:

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.

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

If det(A)=3 and det(B)=2, then det(\(A\)^T\)\(B\)^T) equals:

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

For each real \(x\), \(-1

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

If $A^T [A|B = A|B]B^T$, then find $|A|$

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

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

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19}(\text{adj } A) + \dots + A(\text{adj } A)^{19} + (\text{adj } A)^{20} \text{ is equal to}

Let A be a 3x3 matrix such that A^2 = A. If det(A) = 0 and the trace of A is 2, then the rank of A is:

Let f(x) = 1+sin2xcos2x4sin2xsin2x1+cos2x4sin2xsin2xcos2x1+4sin2x, then the maximum value of f(x), is-

If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix}$, $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{x}(A^2 + cA + d)$ then the sum of values of $c$ and $d$ is

If \(\det(A)\)=4, then det(A\)^6A^3A⁻^3) equals:

If \det(A) = -2, then \det((-\(A)^3) is:

If a determinant has a row of zeros, then its value is:

If $A$ is a square matrix of order $n \times n$ and $k$ is a scalar, then $adj(kA)$ is equal to

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: