Question

Let $A$ be a square matrix of order $3$ whose all entries are $1$ and let $I_3$ be the identity matrix of order $3$. Then the matrix $A-3I_3$ is

• A. invertible
• B. orthogonal
• C. non-invertible
• D. real skew symmetric matrix

Answer 1

The correct option is C non-invertible

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right],I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$A-3I_3=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$

$det(A-3I_3)=-2\left(3\right)-1(-3)+1\left(3\right)=0$
$\Rightarrow A-3I_3$ is non invertible.