Matrices & Determinants Questions (2045)

If \(\ell will always be greater than -

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 -

Let d ∈ R, and A = -24+dsinθ-21sinθ+2d52sinθ-d-sinθ+2+2d, θ ∈ [0, 2π]. If the minimum value of det(A) is 8, then a value of d is :

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.

Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is

The number of triplets (α, β, γ) satisfying the following constraints2α - β + 3γ = 4α + β - 3γ = -15α - β + 3γ = 7αβγ ≤ 0& α, β, γ ∈ I

Let a determinant is given by A = abcpqrxyz and suppose A = 6. If B = p+xq+yr+za+xb+yc+za+pb+qc+r then

If A = ete-t cos te-t sin tet-e-t cos t - e-t sin t-e-t sin t + e-t cos tet2e-t sin t-2e-t cos t Then A is -

For the matrices $A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix}$ and $B=\begin{bmatrix}-29&49\\-13&18\end{bmatrix}$, if $(A^{15}+B)\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$, then among the following which one is true?

If A-1 = 1-100-2100-1, then

Let $|A|=6$, where $A$ is a $3\times3$ matrix. If $|\text{adj}(3\,\text{adj}(A^2\cdot\text{adj}(2A)))|=2^m\cdot3^n$, $m,n\in\mathbb{N}$, then $m+n$ is equal to _____.

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 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-1)(x-2) & Dr = 1102r70173r+1111. The value of f(50) - D5 is -

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

Let $A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix}$ and $B$ be two matrices such that $A^{100}=100B+I$. Then the sum of all the elements of $B^{100}$ is _____.

If A = $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ & A$^n$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, (where n $\ge$ 2 & n $\in$ N), then -

If a, b, c are sides of a scalene triangle, then the value of abcbcacab is :

The value of k for which the set of equations 3x + ky - 2z = 0, x + ky + 3z = 0 and 2x + 3y - 4z = 0 has a non-trivial solution is-

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let E = 12323481318, P = 100001010 and F = 132132243.If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE?

If A is skew symmetric matrix of order 3 and X be another matrix of same order, then |XA + AXT| is (where |P| denotes determinant of matrix P) -

If A & B are two non singular matrices of order 3 × 3 such that AT + B = I & BAT = -B, then which is/are always true (where XT denotes transpose of X and I denotes unit matrix)-

Column-I(A) Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form 1abω1cω2ω1, where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is-(B) Let M be 3 × 3 matrix satisfying M100=-123,M1-10=11-1 and M111=0012. Then the sum of the diagonal entries of M is(C) The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system Axyz=100 has exactly two distinct solutions, is(D) Let k be a positive real number and let A=2k-12k2k2k1-2k-2k2k-1 and B=02k-1k1-2k02k-k-2k0. If det(adj A) + det(adj B) = 106, then [k] is equal to [Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].Column-II(P) 0(Q) 4(R) 9(S) 2

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

Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right) \neq 0, \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Then, among the statements: (I) Trace $(R) = 0$ (II) If trace $(\text{adj}(\text{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.

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:

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Let $M=(a_{ij}), i,j \in \{1, 2, 3\}$, be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j+1$ is divisible by $i$, otherwise $a_{ij} = 0$. Then which of the following statements is (are) true ?(A) $M$ is invertible(B) There exists a nonzero column matrix $\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ such that $M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} -a_1 \\ -a_2 \\ -a_3 \end{pmatrix}$(C) The set $\{X \in \mathbb{R}^3 : MX=0\} \neq \{0\}$, where $0 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$(D) The matrix $(M - 2I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix

If M = 5/23/2-3/21/2, then which of the following matrices is equal to M2022?

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

Let $A=\begin{bmatrix}0&2&-3\\-2&0&1\\3&-1&0\end{bmatrix}$ and $B$ be a matrix such that $B(I-A)=I+A$. Then the sum of the diagonal elements of $B^TB$ is equal to _____.

74 (B): Let a, b, c ∈ ℝ+ and the system of equations\[(1-a)x + y + z = 0\]\[x + (1-b)y + z = 0\]\[x + y + (1-c)z = 0\]has infinitely many solutions. If λ be the minimum value of abc, then λ is divisible by

Which of the following values of α satisfy the equation (1+α)2(1+2α)2(1+3α)2(2+α)2(2+2α)2(2+3α)2(3+α)2(3+2α)2(3+3α)2 = -648α?

If a, b, c are in A.P. and α, β, γ are positive real numbers in G.P., then the equation x+ax2+logαkx+bx2+logβkx+cx2+logγk=0 :-

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

The set of all values of λ for which the system of linear equations :2x1 - 2x2 + x3 = λx1, 2x1 - 3x2 + 2x3 = λx2, -x1 + 2x2 = λx3has a non-trivial solution

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 be a 2 x 2 matrix with det (A) = -1 and det ((A + I) (Adj (1) + I)) = 4. Then the sum of the diagonal elements of A can be :

For some $\alpha,\beta\in\mathbb{R}$, let $A=\begin{bmatrix}\alpha&2\\1&2\end{bmatrix}$ and $B=\begin{bmatrix}1&1\\1&\beta\end{bmatrix}$ be such that $A^2-4A+2I=B^2-3B+I=O$. Then $\left(\det\!\left(\text{adj}\!\left(A^3-B^3\right)\right)\right)^2$ is equal to _____.

If A and B are symmetric matrices and AB = BA, then A-1B is a -

Let B = $$\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$$ and A be a 2 x 2 matrix such that AB^{-1} = A^{-1}. If BCB^{-1} = A and C^4 + \alpha C^2 + \beta I = 0, then 2\beta - \alpha is equal to :

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Let α and β be the distinct roots of the equation x2 + x - 1 = 0. Consider the set T = {1, α, β}. For a 3 × 3 matrix M = (aij)3×3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3. Match each entry in List-I to the correct entry in List-II.List-I(P) The number of matrices M = (aij)3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is(Q) The number of symmetric matrices M = (aij)3×3 with all entries in T such that Cj = 0 for all j, is(R) Let M = (aij)3×3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set { (x, y, z) ∈ R3 : M(x, y, z)T = 0 } is(S) Let M = (aij)3×3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M isList-II(1) 1(2) 12(3) infinite(4) 6(5) 0

Let A, B and A + B are non-singular matrices of order 3 x 3 satisfying A^-1 + B^-1 = (A + B)^-1 and |AB^-1| is R then value of |A|/|B| is

If α, β, γ satisfy the equation <mfenced open="|

If the equations a(y + z) = x, b(z + x) = y, c(x + y) = z (where a, b, c ≠ -1) have nontrivial solutions, then find the value of 1/(1+a) + 1/(1+b) + 1/(1+c).

If Δ(x) = 02x-22x+8x-14x2+700x+4 and f(x) = ∑i=13∑j=13aijcij, where a_{ij} is the element of i^th row and j^th column in Δ(x) and c_{ij} is the cofactor of a_{ij} ∀ i and j, then find the greatest value of f(x), where x ∈ [-3, 18].

Let A and B be two symmetric matrices of order 3.Statement-1 : A(BA) and (AB)A are symmetric matrices.Statement-2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.

Let p, q, r be nonzero real numbers that are, respectively, the 10th, 100th and 1000th terms of a harmonic progression. Consider the system of linear equationsx + y + z = 110x + 100y + 1000z = 0qrx + pry + pqz = 0Match List-I with List-II.

If A and B are two 3 x 3 matrices such that their product AB is a null matrix then is/are always true (where XT denotes transpose of X and I denotes unit matrix)-(A) det. A ≠ 0 ⇒ B must be a null matrix.(B) det. B ≠ 0 ⇒ A must be a null matrix.(C) If none of A and B are null matrices then atleast one of the two matrices must be singular.(D) If neither det. A nor det. B is zero then the given statement is not possible.