Matrices & Determinants Questions - JEE

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:

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 -

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-

Matrices & Determinants Questions (2045)