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A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0$ denotes the $2\times 2$ matrix consisting of zeros):

1. $A^TA$
2. $\begin{bmatrix} 0&A^T \\ A & 0 \end{bmatrix}$
3. $AA^T$
4. $\begin{bmatrix} A & 0 \\ 0 & A^T \end{bmatrix}$

Well, the best way to go is to have a quick look at the options.

The first one says $A^TA$. Let us transpose it. So, $(A^TA)^T = (A)^T *(A^T)^T = A^TA$

So, option s stands true for the symmetricity principle.

In the answer key they have mentioend that (a), (b) and © are correct, can you please explain how (b) is correct and (d) is not?
Yes I have the same issue. I think the answer might be wrong. It should be a, c and d.