Matrices and De...

The system of equations 3χ - y + 4z = 3, χ + 2y - 3z = -2, 6χ + 5y + λz = -3 and has atleast one solutions, if

The values of a for which the system of equations χ + y + z = 0, χ + ay + az = 0 and χ - ay + z = 0 possesses non - zero solutions, are given by

If the homogeneous system of linear equations pχ + y + z = 0, χ + qy + z = 0 and χ + y + rz = 0, where p, q, r ≠ 1, have a non - zero solution, then the value of

If a system of the equations (α + 1)³χ + (α + 2)³y - (α + 3)³ = 0, (α + 1)χ + (α + 2)y - (α + 3) = 0 and χ + y - 1 = 0 is consistent. Then, what is the value of α ?

If X and Y are 2 x 2 matrices such that 2Y + 3Y = 0 and X + 2Y = I, where O and I denote the 2 x 2 zero matrix and the 2 x 2 identity matrix, then X is equal to

The symmetric part of the matrix

If A and B are two symmetric matrices of order 3.
Statement I A(BA) and (AB) A are symmetric matrices.
Statement II AB is symmetric matrix, If matrix multiplication of A and B is commutative.

If A = [a_ij]_2×2, where a_ij = i + j, then A is equal to