Determinants Questions (2072)

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

If \(\det(A)\)=4, then det(A\)⁻^4A^8A⁻^2A) equals:

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

If \(\det(A)\)=4, then det(A\)^5A⁻^5A^3) equals:

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

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

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

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

If \(\det(A)\)=3, then det(A\)^3A^2A⁻^4A^{-1}) equals:

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

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

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

Match the following for the system of linear equationsλx + y + z = 1, x + λy + z = λ, x + y + λz = λ2Column-IColumn-II(A) λ = 1(P) unique solution(B) λ ≠ 1(Q) infinite solutions(C) λ ≠ 1, λ ≠ -2(R) no solution(D) λ = -2(S) finite many solutions

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

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

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

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

The value of α(β² + γ²) + β(γ² + α²) + γ(α² + β²) is divisible by -

If \(\det(A)\)=4, then det(A\)^5(I\) + \(A\)⁻^5)(I\) - \(A\)⁻^5)) equals:

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

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

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 M and N be two 3 x 3 matrices such that MN = NM. Further, if M != N^2 and M^2 = N^4, then

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

Let three matrices be A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}; B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix} and C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}, then tr(A) + tr\left(\frac{ABC}{2}\right) + tr\left(\frac{A(BC)^2}{4}\right) + tr\left(\frac{A(BC)^3}{8}\right) + ....... + \infty is equal to

If $M = \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix}$ and $N = \begin{bmatrix} 0 & 5 \\ 2 & 0 \end{bmatrix}$, then $M^{2011}$ is -

Let three matrices be A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}; B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix} and C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}, then tr(A) + tr\left(\frac{ABC}{2}\right) + tr\left(\frac{A(BC)^2}{4}\right) + tr\left(\frac{A(BC)^3}{8}\right) + ....... + \infty is equal to

If the product of n matrices $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \dots \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ is equal to the matrix $\begin{bmatrix} 1 & 378 \\ 0 & 1 \end{bmatrix}$ then the value of n is equal to -

Prove that $$\begin{vmatrix} a & b & c \\ x & y & z \\ p & q & r \end{vmatrix} = \begin{vmatrix} y & b & q \\ x & a & p \\ z & c & r \end{vmatrix}$$

If $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix}, A^{-1} = \begin{bmatrix} 1/2 & -1/2 & 1/2 \\ -4 & 3 & c \\ 5/2 & -3/2 & 1/2 \end{bmatrix}$, then -

The system of linear equations x + y + z = 6, x + 2y + 3z = 14 and 2x + 5y + pz = q have -

For a real number α, if the system <mfenced open="[

Let $S$ be the set of all $3 \times 3$ symmetric matrices whose entries are either $0$ or $1$. Two of these entries are $1$ and four of them are $0$. A matrix is selected from set $S$, what is the probability that the selected matrix is non singular

System of equation x + y + az = b, 2x + 3y + a^2z = ab + 2 has

Let Dk = <mfenced open="|

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

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 :-

Consider a system of linear equations aix + biy + ciz = di (where ai, bi, ci ≠ 0 and i = 1,2,3 ) & (α,β,γ) is its unique solution, then match list-I with list-IIList-I(I) If ai = di = k2, (k ≠ 0) and α + β + γ = 2, then k is(II) If ai = di = k ≠ 0, then α + β + γ is(III) If ai = k > 0, di = k + 1, then α + β + γ can be(IV) If ai = k di = k + 1, then α + β + γ can beList-II(P) 1(Q) 2(R) 0(S) 3(T) -1

There are two numbers x making the value of the determinant 1-252x-1042x equal to 86. The sum of these two numbers, is-

If a, b, c > 0 and x, y, z ∈ R, then the determinant ax+a-x2ax-a-x21by+b-y2by-b-y21cz+c-z2cz-c-z21 is equal to -

For a determinant Δ of order 3, the element aij is defined as aij = tan-1(tan(i - j)) ∀ i, j, then the value of Δ is equal to (where 'i' represents row and 'j' represents column)

If the determinant a+pl+xu+fb+qm+yv+gc+rn+zw+h splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K, is-

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\[A^{50}U_1 = \begin{bmatrix}1\\25\\25\end{bmatrix},\quad A^{50}U_2 = \begin{bmatrix}0\\1\\0\end{bmatrix},\quad A^{50}U_3 = \begin{bmatrix}0\\0\\1\end{bmatrix}\]The value of \(|U|\) equals

If $A, B$ are two matrices such that $A + B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}, A - B = \begin{bmatrix} 3 & 2 \\ -2 & 0 \end{bmatrix}$, then find $AB$.

If the adjoint of a 3 × 3 matrix P is 144217113, then the possible value(s) of the determinant of P is (are) -

Let M = \begin{bmatrix} a & -360 \\ b & c \end{bmatrix}, where a, b and c are integers. Find the smallest positive value of b such that M^2 = \mathbf{0}, where \mathbf{0} denotes 2 \times 2 null matrix.

Let a1, a2, a3, ..., a10 be in G.P. with ai > 0 for i = 1,2,..., 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which . Then the number of elements in S, is :

Let $A_k=[a_{ij}]$ be square matrix of order 3 with $a_{ij}=(i-j)^k$ for all $i,j\in\{1,2,3\}$. Determinant value of $|A_1+A_3+A_5+\cdots+A_{2023}|$ equals

Let A be a non-singular matrix of order 3. If det(3adj(2adj((detA)A))) = 3-13 ⋅ 2-10 and det(3adj(2A)) = 2m ⋅ 3n, then |3m + 2n| is equal to ____.

Let S be the set of all column matrices \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} such that b_1, b_2, b_3 \in \mathbb{R} and the system of equations (in real variables) -x + 2y + 5z = b_1 2x - 4y + 3z = b_2 x - 2y + 2z = b_3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \in S?