Let `A=``[(0,0,-10),(0,-1,0),(-1,0,0)]` Then only correct statement about the matrix A is (A) A is a zero matrix (B) `A^2=1` (C) `A^-1` does not exist (D) `A=(-1)` I where I is a unit matrix

Options

A. `A^(-1)` does not exist
B. `A=(-1)I`
C. A is a unit matrix
D. `A^(2) = I`

Answer

D

Explanation

Given, `A=[{:(0,0,-1),(0,-1,0),(-1,0,0):}]`
`therefore |A|=[{:(0,0,-1),(0,-1,0),(-1,0,0):}]=-1(-1)=1 ne 0`
`therefore A^(-1)` exists
Now, `A^(2)=[{:(0,0,-1),(0,-1,0),(-1,0,0):}][{:(0,0,-1),(0,-1,0),(-1,0,0):}]=[{:(1,0,0),(0,1,0),(0,0,1):}]`
`rArr A^(2)=I`