`A` and `B` be `3xx3` matrices such that `AB+A=0`, then

Options

A. `(A+B)^(2)=A^(2)+2AB+B^(2)`
B. `|A|=|B|`
C. `A^(2)=B^(2)`
D. none of these

Answer

A

Explanation

`(a)` Given `AB+A+B=0`
`impliesAB+A+B+I=I`
`impliesA(B+I)+(B+I)=(`
`implies(A+I)(B+I)=I`
`implies (A+I)` and `(B+I)` are inverse of each other
`implies(A+I)(B+I)=(B+I)(A+I)`
`impliesAB=BA`
Thus `A` and `B` are commutative
`implies(A+B)^(2)=A^(2)+2AB+B^(2)`