Question

Let $A$ be a symmetric matrix such that $A^4=0$ and $B=I+A+A^2+A^3,$ then $B$ is

• A. singular matrix
• B. symmetric matrix
• C. non-singular matrix
• D. skew symmetric matrix

Answer 1

Answer: (B), (C)

non-singular matrix

$B=I+A+A^2+A^3$
$(I-A)B=(I-A)(I+A+A^2+A^3)$
$=I^2-A^4=I$
$(I-A)B=I$
$B^{-1}=I-A$
$B$ is inverse of $I-A$
$A$ is symmetric$\Rightarrow I-A$ is symmetric
$\Rightarrow$ inverse of $I-A$ is also symmetric
$\Rightarrow B$ is symmetric
$\Rightarrow B$ is non singular