A matrix $A$ is called symmetric if $A=A^{\trans}$.

In this problem, we need the following property of transpose:
Let $A$ be an $m\times n$ and $B$ be an $n \times r$ matrix. Then we have
\[(AB)^{\trans}=B^{\trans}A^{\trans}.\]
(When you distribute transpose over the product of two matrices, then you need to reverse the order of the matrix product.)

Solution.

(a) The product $AB$ is symmetric if and only if $AB=BA$.

Suppose $AB$ is symmetric. This means that we have
\[ AB=(AB)^{\trans}=B^{\trans}A^{\trans}=BA.\]
The second equality is a general property of transpose. In the last equality we used the assumption that $A, B$ are symmetric, that is, $A=A^{\trans}, B=B^{\trans}$. Thus we have $AB=BA$.

On the other hand, if $AB=BA$, then we have
\[AB=BA=B^{\trans}A^{\trans}=(AB)^{\trans}.\]
Thus, we have $AB=(AB)^{\trans}$, hence $AB$ is symmetric.

If the product $AB$ is a diagonal matrix, then $AB=BA$.

Note that a diagonal matrix is symmetric. Hence the result follows from part (a).

Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix
Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\]
That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.
Determine which matrices are in the subspace $W$ […]

If matrix product $AB$ is a square, then is $BA$ a square matrix?
Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix.
Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]

If the Matrix Product $AB=0$, then is $BA=0$ as Well?
Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.
Is it true that the matrix product with opposite order $BA$ is also the zero matrix?
If so, give a proof. If not, give a […]

A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal
Let
\[D=\begin{bmatrix}
d_1 & 0 & \dots & 0 \\
0 &d_2 & \dots & 0 \\
\vdots & & \ddots & \vdots \\
0 & 0 & \dots & d_n
\end{bmatrix}\]
be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.
Let $A=(a_{ij})$ be an $n\times n$ matrix […]

A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues
Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.
Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.
Hint.
Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.
[…]

Questions About the Trace of a Matrix
Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.
(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]

Linear Properties of Matrix Multiplication and the Null Space of a Matrix
Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.
Then find $A\mathbf{w}$.
Hint.
Recall that the null space of an […]

True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$
Let $A$ and $B$ be $2\times 2$ matrices.
Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.
Hint.
In general, matrix multiplication is not commutative: $AB$ and $BA$ might be different.
Solution.
Let us calculate $(A-B)(A+B)$ as […]

For what value(s) of $a$ does the system have nontrivial solutions? \begin{align*} &x_1+2x_2+x_3=0\\ &-x_1-x_2+x_3=0\\ & 3x_1+4x_2+ax_3=0. \end{align*}