The transpose

Let $X=\left[\begin{array}{rrr}-1&-1&1\end{array}\right]$ and $Y=\left[\begin{array}{rrr}0&1&2\end{array}\right]$. Find $X^{T}Y$ and $XY^{T}$ if possible.

Consider the matrices

Find the following if possible. If it is not possible explain why.

1. $-3A{{}^{T}}$.

2. $3B-A^{T}$.

3. $E^{T}B$.

4. $EE^{T}$.

5. $B^{T}B$.

6. $CA^{T}$.

7. $D^{T}BE$.

Which of the following matrices are symmetric, antisymmetric, both, or neither?

Suppose $A$ is a matrix that is both symmetric and antisymmetric. Show that $A=0$.

Let $A$ be an $n\times n$-matrix. Show $A$ equals the sum of a symmetric and an antisymmetric matrix. Hint: Show that $\frac{1}{2}(A^{T}+A)$ is symmetric and then consider using this as one of the matrices.

Show that the main diagonal of every antisymmetric matrix consists of only zeros. Recall that the main diagonal consists of every entry of the matrix which is of the form $a_{ii}$.

Show that for $m\times n$-matrices $A,B$ and scalars $r,s$, the following holds:

Let $A$ be a real $m\times n$-matrix and let $\mathbf{u}\in\mathbb{R}^{n}$ and $\mathbf{v}\in\mathbb{R}^{m}$. Show $(A\mathbf{u})\boldsymbol{\cdot}\mathbf{v}=\mathbf{u}\boldsymbol{\cdot}(A^{T}\mathbf{v})$.

Show that if $A$ is an invertible $n\times n$-matrix, then so is $A^{T}$ and $(A^{T})^{-1}=(A^{-1})^{T}$.

Suppose $A$ is invertible and symmetric. Show that $A^{-1}$ is symmetric.