If A is a nonsingular matrix satisfying `AB-BA=A`, then prove that det. `(B+I)=` det, `(B-I)`.

Explanation

A is nonsingular det `A ne 0`
Given `AB-BA=A`
Hence `AB=A+BA=A(I+B)`
`implies" dte. "A." det." B=" det." A." det." (I+B)`
or det. B = det. (I+B) (as A is nonsingular) (1)
Again `AB-A=BA`
or `A(B-I)=BA`
or (det. A). Det. (B-I) = det. B. det. A
or det. `(B-I)=` dte. (B) (2)
From (1) and (2), dte. `(B-I)=` det. `(B+I)`