Question

If $A$ is a symmetric matrix or skew symmetric matrix, then $A^2$ is

• A. an orthogonal matrix
• B. a symmetric matrix
• C. a unit matrix
• D. a diagonal matrix

Answer 1

The correct option is B a symmetric matrix

A is a symmetric matrix,

then $A_{ij}=A_{ji}$

$A^T=A$

If A is skew symmetric matrix

then

$A_{ij}=-A_{ji}$

$A^T=-A$

$(A^2{)}^T=(A^T{)}^2)=(-A{)}^2=A^2$ ( As $A^T=Aor{A}^T=-A)$

hence $A^2$ is symmetric matrix.