Determinants Questions (2072)

If $\begin{vmatrix}x-4&2x&2x\\2x&x-4&2x\\2x&2x&x-4\end{vmatrix}=(A+Bx)(x-A)^2$, then the ordered pair $(A,B)$ is equal to

If D1 and D2 are two 3 × 3 diagonal matrices where none of the diagonal element is zero, then -

Let det(adj(adjA)) = 14^4 where A = x2-1-1122-11, x ≠ -25/3, then

Consider the following statements.Statement-1 : If $\begin{bmatrix} 3 & -2 \\ 3 & 0 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} y & y \\ x & x \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ 3y & 3y \\ 10 & 10 \end{bmatrix}$, then $2x + 3y = \lambda$.Statement-2 : Given that $\ell + 5 = p + 2m$, where $A$ is a square matrix of order $n$.$\ell$ = maximum number of distinct entries if $A$ is a triangular matrix.$m$ = maximum number of distinct entries if $A$ is a diagonal matrix.$p$ = minimum number of zeroes if $A$ is a triangular matrix.Statement-3 : Let $A$ be the set of all $3 \times 3$ skew symmetric matrices whose entries are either $-1, 0$ or $1$. If there are exactly three $0$'s, three $1$'s and three $(-1)$'s, then number of such matrices is equal to $\mu$.Then, which of the following options is/are correct?

The number of 3 × 3 non-singular matrices with four entries as 1 and all other entries as 0 is

Let $\Delta_1 = \begin{vmatrix} a & b & c \\ e & d & c+d \\ a & b & a+b+c \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} a & b & a+c \\ b & d & b+d \\ a & c & a+d+c \end{vmatrix}$ then the value of $\left|\frac{\Delta_1}{\Delta_2}\right|$, where $b \neq 0$ and $ad \neq bc$ is ____

If \(f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}\), then \(\lim_{x \to 0} \dfrac{f'(x)}{x}\)

Let $$\begin{vmatrix} x^2 + 3x & x - 1 & x + 3 \\ x + 1 & -2x & x - 4 \\ x - 3 & x + 4 & 3x \end{vmatrix} = ax^4 + bx^3 + cx^2 + dx + e$$ be an identity in $x$, then $-\left[\frac{a + b + c + d + e}{a + e}\right]$ is _____ . (where $[.]$ denotes greatest integer function).

The number of A in Tp such that det (A) is not divisible by p is -

If determinant has a triangular form, its value is:

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

The value of the determinant of a 3x3 matrix is given by the expression. If the matrix is singular, what is the value of the determinant?

If \(A\), \(B\), \(A+I\), \(A+B\) are idempotent matrices, then \(AB\) is equal to

An invertible matrix A of order 3 satisfies the relation A = A-1 + 2I, (where I denotes identity matrix). The value of |A - I|.|A + I|.|A - 2I| is

Let \( A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix} \). If \( u_1 \) and \( u_2 \) are column matrices such that \( Au_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \) and \( Au_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \), then \( u_1 + u_2 \) is equal to

Let \(A = \begin{bmatrix}1 & -1 & 1\\ 2 & 1 & -3\\ 1 & 1 & 1\end{bmatrix}\) and \(10B = \begin{bmatrix}4 & 2 & 2\\ -5 & 0 & \alpha\\ 1 & -2 & 3\end{bmatrix}\). If \(B\) is the inverse of \(A\), then find the value of \(\alpha\).

Let A be a 3x3 matrix such that A^2 - 5A + 7I = 0. If A^4 = aA + bI, then the value of a + b is:

Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 and b3 respectively. If x1 = <mfenced open="[

Let A be a 3 x 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 -> 2R2 + 5R3 on 2A, then det(B) is equal to :

Evaluate \[\begin{vmatrix} \cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha \\ -\sin\beta & \cos\beta & 0 \\ \sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha \end{vmatrix}\]

If $A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$, $A^{-1} = \alpha A + \beta I$ and $\alpha + \beta = -2$, then $4\alpha^2 + \beta^2 + \lambda^2$ is equal to:

Let a and b be two real numbers such that \(a > 1\), \(b > 1\). If \(A = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\), then \(\lim_{n \to \infty} (A^n)^{-1}\) is

If determinant is zero but matrix not zero matrix, rank is:

The determinant \(\begin{vmatrix} y^2 & -xy & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix}\) is equal to

The number of values of k for which the linear equations\(4x + ky + 2z = 0\)\(kx + 4y + z = 0\)\(2x + 2y + z = 0\)possess a non-zero solution is

Let A be a 3x3 matrix and det(A) = 2. If n = det(adj(adj(.....(adj(A))))), where adj is applied 2024 times, then the remainder when n is divided by 9 is equal to ________.

If determinant is zero, then system of linear equations is:

In a $\triangle ABC$, if $$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix} = 0$$, then $\sin^2 A + \sin^2 B + \sin^2 C = $_____.

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

Let $M = \begin{bmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{bmatrix}$ and $adjM = \begin{bmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?(A) $a + b = 3$(B) $\det(adjM^2) = 81$(C) $(adjM)^{-1} + adjM^{-1} = -M$(D) If $M \begin{bmatrix} \alpha \\ 1 \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $\alpha - \beta + \gamma = 3$

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

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

For Problems 1–3Let \(A\) be a matrix of order \(2 \times 2\) such that \(A^2 = O\).\(\text{tr}(A)\) is equal to

Let β be a real number. Consider the matrixA = β0121-231-2If A7 - (β-1)A6 - βA5 is a singular matrix, then the value of 9β is ____.

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

If \(A\) and \(B\) are square matrices of order 3 such that \(\det(A) = -2\) and \(\det(B) = 1\), then find the value of \(\det\left(A^{-1} \cdot \text{adj}(B^{-1}) \cdot \text{adj}(2A^{-1})\right)\).

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

If \( A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix} \), then the value of \( \alpha \) for which \( A^2 = B \) is

The value of the determinant of a matrix is given by the answer key.

Let A = [aij] be a 3x3 matrix such that aij = 2i-j for all i, j. Then the matrix An for any positive integer n is equal to:

Let A be a 3x3 matrix such that A^2 - 5A + 7I = 0. If A^n = 5^n A - 7^n I for some n, then n is equal to:

Let A be a 3x3 matrix such that A^2 = I. If the determinant of A is -1, then the trace of A is:

Let $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ & $P = \begin{bmatrix} \cos \frac{\pi}{12} & \sin \frac{\pi}{12} \\ -\sin \frac{\pi}{12} & \cos \frac{\pi}{12} \end{bmatrix}$ and $Q = P^T AP$, then if $PQ^{2014}P^T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ then sum of digits of $b$ is _____.

Let A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}. If M and N are two matrices given by M = \sum_{k=1}^{10} A^{2k} and N = \sum_{k=1}^{10} A^{2k-1} then MN^2 is (1) a non-identity symmetric matrix (2) a skew-symmetric matrix (3) neither symmetric nor skew-symmetric (4) an identity matrix

If \(A = \begin{bmatrix}i & -i\\ -i & i\end{bmatrix}\) and \(B = \begin{bmatrix}1 & -1\\ -1 & 1\end{bmatrix}\), then \(A^8\) equals

The value of an odd order determinant in which aij + aji = 0 ∀ i, j is -

If $S_r = \begin{vmatrix} 2r & x & n(n+1) \\ 6r^2-1 & y & n^2(2n+3) \\ 4r^3-2nr & z & n^3(n+1) \end{vmatrix}$, then $\sum_{r=1}^n S_r$ does not depend on -

If a, b, c are sides of a scalene triangle, then the value of <mfenced close="|

If S is the set of distinct values of b for which the following system of linear equations has no solution, then S is\[x + y + z = 1\]\[x + ay + z = 1\]\[ax + by + z = 0\]

The determinant <mfenced close="|