Let A and b are two square idempotent matrices such that `ABpm BA` is a null matrix, the value of det (A - B)
cannot be equal

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

A, B, C

Explanation

`because (A-B)^(2) = A^(2)- AB - BA + B^(2)`
`= A+ B [because AB + BA = 0 and A^(2) = A, B^(2) = B]`
`therefore abs(A-B)^(2) = abs (A+B)` ... (i)
and `(A+B)^(2) =
A^(2) + AB + BA + B^(2) `
`= A+B [ because AB + BA = 0 and A^(2) = A, B^(2) = B]`
`rArr abs(A+B)^(2)= abs(A+B)`
` rArr abs(A+B) (abs(A+B)-1) = 0`
`therefore abs(A+B) = 0, 1`
From E. (i),
`abs(A-B)^(2) = 0, 1 rArr abs(A-B )= 0 pm 1 `
or det `(A-B) = 0, -1, 1`