`[{:(0,0,0),(0,0,0):}]ne[{:(0,0),(0,0),(0,0):}].` Why ?

If P=[{:(x,0,0),(0,y,0),(0,0,z):}]andQ=[{:(a,0,0),(0,b,0),(0,0,c):}] , prove that PQ=[{:(xa,0,0),(0,yb,0),(0,0,zc):}]=QP .

If P=[{:(x,0,0),(0,y,0),(0,0,z):}] and Q=[{:(a,0,0),(0,b,0),(0,0 ,c):}] then prove that PQ=[{:(xa,0,0),(0,yb,0),(0,0,zc):}]=QP .

Find a matrix B such that B[(a,0,0),(0,a,0),(0,0,a)]=[(1,0,0),(0,1,0),(0,0,1)], a ne 0

Out of the following matrices, choose that matrix which is a scalar matrix: a) [(0 ,0),( 0, 0)] (b) [(0, 0, 0),( 0 ,0, 0)] (c) [(0, 0),( 0, 0),( 0 ,0)] (d) [(0),(0),( 0) ]

If P = [(x,0,0),(0,y,0),(0,0,z)]and Q = [(a,0,0),(0,b,0),(0,0,c)] , verify that PQ = QP = [(xa, 0,0),(0,yb,0),(0,0,zc)]

If A=[(0,0,0,0),(0,0,0,0),(1,0,0,0),(0,1,0,0)] then (A) A^2=I (B) A^2=0 (C) A^3=0 (D) none of these

If A=[(0,0,0,0),(0,0,0,0),(1,0,0,0),(0,1,0,0)] then (A) A^2=I (B) A^2=0 (C) A^3=0 (D) none of these

The inverse of the matrix [(0,0,1),(0,1,0),(1,0,0)] is (A) [(0,0,1),(0,1,0),(1,0,0)] (B) [(0,0,-1),(0,-1,0),(-1,0,0)] (C) [(1,0,0),(0,1,0),(0,0,1)] (D) [(1/2,0,0),(0,1/2,0),(0,0,1/2)]

The inverse of the matrix [(0,0,1),(0,1,0),(1,0,0)] is (A) [(0,0,1),(0,1,0),(1,0,0)] (B) [(0,0,-1),(0,-1,0),(-1,0,0)] (C) [(1,0,0),(0,1,0),(0,0,1)] (D) [(1/2,0,0),(0,1/2,0),(0,0,1/2)]

Compute [(a,0),(a,0)][(0,0),(0,0)] .