Question

Which of the following statements is/are true about square matrix A of order n?

• A. $(-A{)}^{-1}$ is equal to $(-A{)}^{-1}$ when n is odd only.
• B. If $A^n=0$, then $I+A+A^2+\dots \dots A^{n-1}=(I-A{)}^{-1}$
  
• C. If A is skew-symmetric matrix of odd order, then its inverse does not exist.
• D. $(A^T{)}^{-1}=(A^{-1}{)}^T$ holds always

Answer 1

Answer: (B), (C)

The correct options are
B If $A^n=0$, then $I+A+A^2+\dots \dots A^{n-1}=(I-A{)}^{-1}$

C If A is skew-symmetric matrix of odd order, then its inverse does not exist.

$(-A{)}^{-1}=\frac{adj(-A)}{|-A|}=\frac{(-1{)}^{n-1}adjA}{(-1{)}^n|A|}=\frac{adj\left(A\right)}{-|A|}=-A^{-1}$
$If{A}^n=0$
$(I-A)(I+A+A^2+\dots \dots A^{n-1})=I-A^n=1$
$\Rightarrow (I-A{)}^{-1}=I+A+A^2+\dots \dots A^{n-1}$