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, \(a = -1,\ b = 0\)
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
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