Determinants Questions (2072)

If the matrix \(A = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}\) where \(a, b, c\) are positive real numbers such that \(abc = 1\) and \(A^T A = I\), then find the value of \(a^3 + b^3 + c^3\).

Let A = $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{bmatrix}$$. Then the sum of the diagonal elements of the matrix $$(A + I)^{11}$$ is equal to:

Let P be a 2 x 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 p_1 and p_2 (p_1 > p_2) are two values of p for which det(P - pI) = 0, where I is an identity matrix of order 2, then (5p_1 + 2p_2) is equal to [Note : det(M) denotes determinant of square matrix M]

If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then xzy2 is equal to :

Let P be a 2 x 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 p_1 and p_2 (p_1 > p_2) are two values of p for which det(P - pI) = 0, where I is an identity matrix of order 2, then (5p_1 + 2p_2) is equal to [Note : det(M) denotes determinant of square matrix M]

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

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

The determinant <mfenced close="|

Let f(x)=1x-1x2(x-1)(x-1)(x-2)x(x-1)3(x-1)(x-2)(x-1)(x-2)(x-3)x(x-1)(x-2) & Dr=1102r70173r+1111. The value of ∑r=110Dr is

System of linear equations in x, y, z have infinite solutions which2x + y + z = 1x - 2y + z = 23x - y + 2z = 3

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

Let $D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}$ and $D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}$ then the value of $\frac{D_1}{D_2}$ where $b \neq 0$ and $ad \neq bc$, is

If \(\det(A)\)=5, then det((A\)^{-1})^2) equals:

Let $\alpha\beta\neq0$ and $A=\begin{bmatrix}\beta&\alpha&3\\\alpha&\alpha&\beta\\-\beta&\alpha&2\alpha\end{bmatrix}$. If $B=\begin{bmatrix}3\alpha&-9&3\alpha\\-\alpha&7&-2\alpha\\-2\alpha&5&-2\beta\end{bmatrix}$ is the matrix of cofactors of the elements of $A$, then $\det(AB)$ is equal to:

If A is a square matrix of order 3 such that det(A) = 3 and det(adj(-4 adj(-3 adj(3 adj((2A)^(-1)))))) = 2^m 3^n, then m + 2n is equal to:

Let $A=\begin{bmatrix}2&a&0\\1&3&1\\0&5&b\end{bmatrix}$. If $A^3=4A^2-A-21I$, where $I$ is the identity matrix of order $3\times3$, then $2a+3b$ is equal to:

If \(\det(A)\)=3, then det(A\)^5A⁻^3A^{-1}) equals:

If \det(A)=k, then \(\det(kA)\) for \(2 \times 2\) matrix is:

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

If \(\det(A)\)=5, then det(A\)⁻^2A^4A^{-1}) equals:

If \(\det(A)\)=2, then det(A\)^2A^3A⁻^4) equals:

If A is a non-singular matrix and AT denotes the transpose of A, then :

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

If \(\det(A)\)=4, then det(A\)^4A^2A⁻^5) equals:

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19} + ... + A + I = ?

If \(\det(A)\)=3, then det(A\)^3A^2A⁻^4) equals:

If $A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$ and $(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, then $13(a^2 + b^2)$ is equal to ____.

If \(\det(A)\)=4, then det(A\)^5A^5A⁻^9) 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 the system of equation, a2x - ay = 1 - a & bx + (3 - 2b)y = 3 + a possess a unique solution x = 1, y = 1 than :

If the system of linear equations x1 + 2x2 + 3x3 = 6 x1 + 3x2 + 5x3 = 9 2x1 + 5x2 + ax3 = b is consistent and has infinite number of solutions, then :-

Which of the following is an orthogonal matrix -

The set of all values of λ for which the system of linear equations 2x1 - 2x2 + x3 = λx1, 2x1 - 3x2 + 2x3 = λx2, -x1 + 2x2 = λx3 has a non-trivial solution

The determinant <mfenced open="|

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

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

If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then xzy2 is equal to :

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

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

If \(\det(A)\)=4, then det(A\)^5A^2A⁻^5) equals:

If P is a 3 × 3 real matrix such that PT = aP + (a-1)I, where a > 1, then

If \(\det(A)\)=5, then det(A\)^6A^3A⁻^6) equals:

If \(\det(A)\)=k, then det(A\)^nA^mA⁻^n) where m>0 equals:

If \(\det(A)\)=5, then det(A\)^7A^4A⁻^4A⁻^7A^5A⁻^5) 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 \(\det(A)\)=4, then det((A\)^5A⁻^5)^6) equals:

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

If A and B are two orthogonal matrices of order 3, then -(A) A and B both will be invertible matrices(B) matrix ABA will also be orthogonal(C) matrix A^2B^2 will also be orthogonal(D) maximum value of det\left(\frac{A}{2} adj(2B)\right) is 8.

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

Let A = 12322-130k and f(x) = x3 - 2x2 - αx + β = 0. If A satisfies f(x) = 0, then-