Matrices & Determinants Questions (2045)

If $ax + by + cz = 0$, $bx + cy + az = 0$, $cx + ay + bz = 0$, $a,b,c \in \mathbb{R}^+$ then:

If $x^a y^b = e^m$, $x^c y^d = e^n$, $P = \begin{pmatrix} m & b \\ n & d \end{pmatrix}$, $Q = \begin{pmatrix} a & m \\ c & n \end{pmatrix}$, $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then:

Let $f(n) = \begin{vmatrix} ^n P_n & ^{n+1} P_{n+1} & ^{n+2} P_{n+2} \\ ^n C_n & ^{n+1} C_{n+1} & ^{n+2} C_{n+2} \end{vmatrix}$, where the symbols have their usual meanings. Then $f(n)$ is divisible by

If $A(\theta) = \begin{pmatrix} \sin \theta & i \cos \theta \\ i \cos \theta & \sin \theta \end{pmatrix}$, then which of the following is not true

If $A = \begin{vmatrix} \sin\theta\cos\theta & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi\sin\phi & \sin\theta\sin\phi & 0 \end{vmatrix}$ then

If $\begin{vmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ca-b^2 \end{vmatrix} = \begin{vmatrix} a^2 & b^2 & b^2 \\ b^2 & a^2 & b^2 \\ b^2 & b^2 & a^2 \end{vmatrix}$, then

$\Delta = \begin{vmatrix} a & a^2 & 0 \\ 1 & 2a+b & a+b \\ 0 & 1 & 2a+3b \end{vmatrix}$ is divisible by

Which of the following values of $a$ satisfy the equation $\begin{vmatrix} (1+a)^2 & (1+2a)^2 & (1+3a)^2 \\ (2+a)^2 & (2+2a)^2 & (2+3a)^2 \\ (3+a)^2 & (3+2a)^2 & (3+3a)^2 \end{vmatrix} = -684a$

If $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is an orthogonal matrix of order 3, then:

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = \left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $P^2 \neq 0$ when $n =$

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, non-zero symmetric matrix. Then which of the following is (are) skew-symmetric:

Which of the following is (are) not the square of a $3 \times 3$ matrix with real entries:

If determinant is triangular, rank is:

Let $\{\Delta_1, \Delta_2, \Delta_3, \ldots \Delta_k\}$ be the set of third order determinants that can be made with the distinct non-zero real numbers $a_1, a_2, \ldots a_9$. Then:

The system of equations $6x + 5y + \lambda z = 0, 3x - y + 4z = 0, x + 2y - 3z = 0$ has:

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

$\begin{bmatrix} 1 & -\tan \theta/2 \\ \tan \theta/2 & 1 \end{bmatrix} \begin{bmatrix} 1 & \tan \theta/2 \\ -\tan \theta/2 & 1 \end{bmatrix}^{-1}$ is equal to -

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

For Problems 9–11Let \(A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}\) satisfies \(A^n = A^{n-2} + A^2 - I\) for \(n \geq 3\). And trace of a square matrix \(X\) is equal to the sum of elements in its principal diagonal.Further consider a matrix \(U_{3\times 3}\) with its columns as \(U_1, U_2, U_3\) such that\[A^{50}U_1 = \begin{bmatrix}1\\25\\25\end{bmatrix},\quad A^{50}U_2 = \begin{bmatrix}0\\1\\0\end{bmatrix},\quad A^{50}U_3 = \begin{bmatrix}0\\0\\1\end{bmatrix}\]Trace of \(A^{50}\) equals

Find the value of $x, y, z$ and $w$ which satisfy the matrix equation $\begin{bmatrix} x+3 & 2y+x \\ z-1 & 4w-8 \end{bmatrix} = \begin{bmatrix} -x-1 & 0 \\ 3 & 2w \end{bmatrix}$.

The system of homogeneous equations $\lambda x + (\lambda + 1)y + (\lambda - 1)z = 0$, $(\lambda + 1)x + \lambda y + (\lambda + 2)z = 0$, $(\lambda - 1)x + (\lambda + 2)y + \lambda z = 0$ has non trivial solution for:

If $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $P = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$, $Q = P^T AP$, then $PQ^{2014}P^T =$

Let $A$ be a $3 \times 3$ non-singular matrix then which of the following is not true

If $A$ be $3 \times 3$ non-singular matrix, $|A| = K$, then $|(xA)^{-1}| = $ (where $x \neq 0$)

Number of distinct real values of $K$, such that the system of equations $x+2y+z=1$, $x+3y+4z=K$, $x+5y+10z=K^2$ has infinitely many solutions is:

$a+d=$

$\frac{c}{b}=$

If $ax + by + cz = 0$, $bx + cy + az = 0$, $cx + ay + bz = 0$, $a,b,c \in \mathbb{R}^+$ then:

Let \(A\) be a \(2 \times 2\) matrix.Statement-1: adj(adj \(A\)) = \(A\)Statement-2: |adj \(A\)| = |\(A\)|

A be the set of all square matrices of order 3 with elements either 0, 1, or $-1$, then:

If \(\Delta\) = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and two rows are swapped twice, determinant becomes:

Let A be the set of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ c & a \end{pmatrix}$, such that $a,b,c \in \{0,1,2,3,4\}$ then:

If $x^a y^b = e^m$, $x^c y^d = e^n$, $P = \begin{pmatrix} m & b \\ n & d \end{pmatrix}$, $Q = \begin{pmatrix} a & m \\ c & n \end{pmatrix}$, $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then:

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

$A = \begin{pmatrix} -3 & -1 & 2 \\ 3 & 1 & -1 \\ 4 & 2 & 5 \end{pmatrix}$, $A \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $A \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$, $A \begin{pmatrix} x_3 \\ y_3 \\ z_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, $B = \begin{pmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{pmatrix}$, then:

Let $f(n) = \begin{vmatrix} ^n P_n & ^{n+1} P_{n+1} & ^{n+2} P_{n+2} \\ ^n C_n & ^{n+1} C_{n+1} & ^{n+2} C_{n+2} \end{vmatrix}$, where the symbols have their usual meanings. Then $f(n)$ is divisible by

If $A(\theta) = \begin{pmatrix} \sin \theta & i \cos \theta \\ i \cos \theta & \sin \theta \end{pmatrix}$, then which of the following is not true

If $A = \begin{vmatrix} \sin\theta\cos\theta & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi\sin\phi & \sin\theta\sin\phi & 0 \end{vmatrix}$ then

If $\begin{vmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ca-b^2 \end{vmatrix} = \begin{vmatrix} a^2 & b^2 & b^2 \\ b^2 & a^2 & b^2 \\ b^2 & b^2 & a^2 \end{vmatrix}$, then

If \(A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}\) such that \(A^2 = B\) then

Let $a, \lambda, \mu \in \mathbb{R}$ consider the system of linear equations $ax + 2y = \lambda$, $3x - 2y = \mu$. Which of the following statement(s) is(are) correct?

$\Delta = \begin{vmatrix} a & a^2 & 0 \\ 1 & 2a+b & a+b \\ 0 & 1 & 2a+3b \end{vmatrix}$ is divisible by

If determinant has two rows proportional, then determinant is:

Let \(A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}\). If \(A^2 = B\), then find \(\alpha\).

If determinant is non-zero, system of equations is:

Which of the following values of $a$ satisfy the equation $\begin{vmatrix} (1+a)^2 & (1+2a)^2 & (1+3a)^2 \\ (2+a)^2 & (2+2a)^2 & (2+3a)^2 \\ (3+a)^2 & (3+2a)^2 & (3+3a)^2 \end{vmatrix} = -684a$

If $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is an orthogonal matrix of order 3, then:

How many $3\times 3$ matrices $M$ with entries from $\{0, 1, 2\}$ are there for which the sum of the diagonal entries of $M'M$ is 5.

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

For Problems 16–18Consider the polynomial function\[f(x) = \begin{vmatrix} (1+x)^a & (1+2x)^b & 1 \\ 1 & (1+x)^a & (1+2x)^b \\ (1+2x)^b & 1 & (1+x)^a \end{vmatrix}\]\(a, b\) being positive integers.Which of the following is true?