Determinants Questions (2072)

$(I + A)^{100} =$

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

A square matrix $P$ satisfies $P^2 = I - P$, where $I$ is the identity matrix. If $P^n = 5I - 8P (n \in N)$, then minimum value of $n$ is equal to

3. If px^4 + qx^3 + rx^2 + sx + t = x2+3xx-1x+3x+12-xx-3x-3x+43x then t is equal to -

Let $A = \begin{bmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{bmatrix}$. If $AA^T = I^3$, then $|p|$ is :

If two rows of a determinant are interchanged, then determinant becomes:

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

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (adj A)^{20} is equal to

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

If \(\det(A)\)=k, then det(I\)\cdotA\cdotI) equals:

If \(\det(A)\)=k, then det((A\)^0)^{-1}) equals:

If \(\det(A)\)=k, then det((A\)^0)^n) equals:

If A = 5!6!7!6!7!8!7!8!9!, then |adj(adj(2A))| is equal to :

Let $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. If $B = \begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$, then the sum of all the elements of the matrix $\sum_{n=1}^{50} B^n$ is equal to

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

Find the sum of all positive integral values of a for which every solution to the system of equation x + ay = 3 and ax + 4y = 6 satisfy the inequalities x > 1, y > 0.

If determinant is expanded along any row, result is:

Let a, b, c are the solutions of the cubic x3 - 5x2 + 3x - 1 = 0, then find the value of the determinant

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$, then $3a^2 + 4b^2$ is equal to

If x + 3 2y + x z - 1 4w - 8 = -x - 1 0 3 2w, then find the value of \(x + y + z + w\).

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

Matrices A and B satisfy \(AB = B^{-1}\), where \(B = \begin{pmatrix} 2 & -2 \\ -1 & 0 \end{pmatrix}\). Find the value of \(\lambda\) for which \(\lambda A - 2B + I = O\), without finding \(B^{-1}\).

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

The total number of matrices A = 02y12xy-12x-y1, (x, y ∈ R, x ≠ y) for which AT A = 3I3 is :-

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

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

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

If $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$, show that $A^k = \begin{bmatrix} 1+2k & -4k \\ k & 1-2k \end{bmatrix}$, where $k$ is any positive integer.

Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.List-I(A) The number of matrices in A is(B) The number of matrices in A for which the system of linear equations Axyz=100 has a unique solution, is(C) The number of matrices in A for which the system of linear equations Axyz=100 is inconsistent, is(D) The number of matrices in A for which the system of linear equations Axyz=100 has infinitely many solutions, isList-II(P) more than 2(Q) 12(R) more than 4(S) less than 7

The value of θ lying between -π/4 & π/2 and 0 ≤ A ≤ π/2 and satisfying the equation 1+sin2 Acos2 A2sin 4θsin2 A1+cos2 A2sin 4θsin2 Acos2 A1+2sin 4θ = 0 are -

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

If a, b, c are in AP, then x+1x+2x+ax+2x+3x+bx+3x+4x+c equals -

If the maximum and minimum values of the determinant $\begin{vmatrix} 1 + \sin^2 x & \cos x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$ are $c$ and $d$ respectively, then $c + 2d$ is equal to

Which of the following is an orthogonal matrix -

The determinant cos(θ+ϕ)-sin(θ+ϕ)cos2ϕsinθcosθsinϕ-cosθsinθcosϕ is -

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

Given system of linear equations:\(x + y + z = 6\) ... (i)\(4x + \lambda y - \lambda z = \lambda - 2\) ... (ii)\(3x + 2y - 4z = -5\) ... (iii)has infinitely many solutions. Find the value of \(\lambda\).

If \(x, y, z\) are integers in AP, lying between 1 and 9, and \(x_{51}, y_{41}\) and \(z_{31}\) are three digit numbers, then the value of \(\begin{vmatrix}x_{51} & y_{41} & z_{31}\\x & y & z\\5 & 4 & 3\end{vmatrix}\) is

Let \(A = [a_{ij}]\) and \(B = [b_{ij}]\) be two \(3 \times 3\) real matrices such that \(b_{ij} = (3)^{i+j-2} a_{ji}\), where \(i, j = 1, 2, 3\). If the determinant of \(B\) is 81, then the determinant of \(A\) is

For the system of linear equation2x - y + 3z = 53x + 2y - z = 74x + 5y + αz = βWhich of the following is(are) CORRECT?

For Problems 19–21Given that the system of equations \(x = cy + bz\), \(y = az + cx\), \(z = bx + ay\) has nonzero solutions and at least one of the \(a, b, c\) is a proper fraction.\(abc\) is

In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & 0 \\ 1 & 2 & 0 & 1 \\ 2 & 7 & \sqrt{3} & \sqrt{5} \end{bmatrix}$(i) The order of the matrix,(ii) The number of elements,(iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.

If \(A = \begin{bmatrix}a & b\\ 0 & a\end{bmatrix}\) is \(n\)th root of \(I_2\), then choose the correct statements:(i) if \(n\) is odd, \(a = 1,\ b = 0\)(ii) if \(n\) is odd, \(a = -1,\ b = 0\)(iii) if \(n\) is even, \(a = 1,\ b = 0\)(iv) if \(n\) is even, \(a = -1,\ b = 0\)

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 -

The value of \(|U^{-1}|\) is

If the system of equation 2x + y - z = 5 2x - 5y + λz = μ x + 2y - 5z = 7 has infinitely many solutions, then (λ + μ)2 + (λ - μ)2 is equal to

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

If Δ = a1b1c1a2b2c2a3b3c3 and A2, B2, C2 are respectively cofactors of a2, b2, c2 then a1A2 + b1B2 + c1C2 is equal to -

If \(A\) is a \(2 \times 2\) matrix and \det(A) = 5, then \det(adj(adj \(A)) is:

If \(A\) is a \(3 \times 3\) matrix and \det(A) = 4, then \det(adj(adj \(A)) is: