Question

Select all statements below which are true for all invertible n×n matrices AA and BB

Answer 1

(a) is false. Consider the identity matrix, $A=I$. Then

$(I+I^{-1})^3=(2I)^3=8I$

bu

$I^3+I^{-1}=I+I=2I$

(b) is false. Since $A$ is invertible, we have

$ABA^{-1}=B\implies (AB)(A^{-1}A)=BA\implies AB=BA$

but in general, matrix multiplication is non-commutative.

(c) is true. Let $B=A^8$; then

$B(A^{-1})^8=A^7(AA^{-1})(A^{-1})^7=A^7(A^{-1})^7$

$B(A^{-1})^8=A^6(AA^{-1})(A^{-1})^6=A^6(A^{-1})^6$

and so on, down to

$B(A^{-1})^8=AA^{-1}=I$

which is to say $B^{-1}=(A^{-1})^8$.

(d) is false. Consider $A=-I$, so that

$(-I)+I=0$

which is singular (i.e. determinant is zero), and thus not invertible.

(e) is true. It follows from the distributivity of matrix multiplication:

$(I+A)(I+A^{-1})=I^2+AI+IA^{-1}+AA^{-1}=I+A+A^{-1}+I=2I+A+A^{-1}$

(f) is false for the same reason as (b). Expanding the product gives

$(A+B)(A-B)=A^2+BA-AB-B^2$

but in general, $AB\neq BA$.