GO Classes DPP | GATE CS | Linear Algebra | Symmetric Matrices

Answer 1

We check each option using the definition of a symmetric matrix: a matrix $M$ is symmetric if $M^T=M$.

For option A, $(A^TA)^T=A^T(A^T)^T=A^TA$, so $A^TA$ is symmetric.

For option B, let $M=\begin{pmatrix}0&A^T\\A&0\end{pmatrix}$. Then

$M^T=\begin{pmatrix}0^T&A^T\\(A^T)^T&0^T\end{pmatrix}=\begin{pmatrix}0&A^T\\A&0\end{pmatrix}=M$.

So this matrix is symmetric.

For option C, $(AA^T)^T=(A^T)^TA^T=AA^T$, so $AA^T$ is symmetric.

For option D, let $N=\begin{pmatrix}A&0\\0&A^T\end{pmatrix}$. Then

$N^T=\begin{pmatrix}A^T&0\\0&A\end{pmatrix}$,

which is not equal to $N$ in general unless $A=A^T$. Since $A$ is arbitrary, this matrix is not always symmetric.

Therefore, the symmetric matrices are $\boxed{A,\ B,\ \text{and}\ C}$.

Answer 2

Use the property :

(AB)^T = B^T A^T

Answer 3

A,B,C

D is not same as M(Tranpose) = M