Question

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

• A. $(-A{)}^{-1}$ is equal to $-A^{-1}$ when n is odd only.
• B. If $A^n=O$, then $I+A+A^2+A^3+\cdots +A^{n-1}=(I-A{)}^{-1}$.
• C. If A is skew-symmetric matrix of odd order, then its inverse does not exist.
• D. If A is invertible matrix, then $(A^T{)}^{-1}=(A^{-1}{)}^T$.

Answer 1

Answer: (B), (C), (D)

The correct options are
B If $A^n=O$, then $I+A+A^2+A^3+\cdots +A^{n-1}=(I-A{)}^{-1}$.
C If A is skew-symmetric matrix of odd order, then its inverse does not exist.
D If A is invertible matrix, then $(A^T{)}^{-1}=(A^{-1}{)}^T$.

$(-A{)}^{-1}=\frac{adj(-A)}{|-A|}=\frac{(-1{)}^{n-1}adjA}{(-1{)}^n|A|}=$
$\frac{adj\left(A\right)}{-|A|}$
$=-A^{-1}$ (for any value of n)
Given $A^n=0$
Now,
$(I-A)(I+A+A^2+\cdots +A^{n-1})=I-A^n=I$
$\Rightarrow (I-A{)}^{-1}=I+A+A^2+\cdots A^{n-1}$
D- part as per the theorem