
# 7.4: Review Problems

1.    Compute the following matrix products
$$\begin{pmatrix}1&2&1\\4&5&2\\7&8&2\end{pmatrix} \begin{pmatrix}-2&\frac{4}{3}&-\frac{1}{3}\\2&-\frac{5}{3}&\frac{2}{3}\\-1&2&-1\end{pmatrix}\, , \qquad \begin{pmatrix}1&2&3&4&5\end{pmatrix} \begin{pmatrix}1\\2\\3\\4\\5\end{pmatrix}\, ,$$
$$\begin{pmatrix}1\\2\\3\\4\\5\end{pmatrix}\begin{pmatrix}1&2&3&4&5\end{pmatrix}\, ,\qquad \begin{pmatrix}1&2&1\\4&5&2\\7&8&2\end{pmatrix} \begin{pmatrix}-2&\frac{4}{3}&-\frac{1}{3}\\2&-\frac{5}{3}&\frac{2}{3}\\-1&2&-1\end{pmatrix} \begin{pmatrix}1&2&1\\4&5&2\\7&8&2\end{pmatrix}\, ,$$
$$\begin{pmatrix}x & y &z\end{pmatrix} \begin{pmatrix} 2& 1& 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix} \begin{pmatrix}x\\y\\z\end{pmatrix}\, ,\qquad \begin{pmatrix}2&1&2&1&2\\0&2&1&2&1\\0&1&2&1&2\\0&2&1&2&1\\0&0&0&0&2\end{pmatrix} \begin{pmatrix}1&2&1&2&1\\0&1&2&1&2\\0&2&1&2&1\\0&1&2&1&2\\0&0&0&0&1\end{pmatrix}\, ,$$

$$\begin{pmatrix}-2&\frac{4}{3}&-\frac{1}{3}\\2&-\frac{5}{3}&\frac{2}{3}\\-1&2&-1\end{pmatrix} \begin{pmatrix}4&\frac{2}{3}&-\frac{2}{3}\\6&\frac{5}{3}&-\frac{2}{3}\\12&-\frac{16}{3}&\frac{10}{3}\end{pmatrix} \begin{pmatrix}1&2&1\\4&5&2\\7&8&2\end{pmatrix}\, .$$

2.    Let's prove the theorem $$(MN)^{T} = N^{T}M^{T}$$.

Note: the following is a common technique for proving matrix identities.

a)    Let $$M=(m^{i}_{j})$$ and let $$N=(n^{i}_{j})$$. Write out a few of the entries of each matrix in the form given at the beginning of section 7.3.

b)    Multiply out $$MN$$ and write out a few of its entries in the same form as in part (a). In terms of the entries of $$M$$ and the entries of $$N$$, what is the entry in row $$i$$ and column $$j$$ of $$MN$$?

c)    Take the transpose $$(MN)^{T}$$ and write out a few of its entries in the same form as in part (a). In terms of the entries of $$M$$ and the entries of $$N$$, what is the entry in row $$i$$ and column $$j$$ of $$(MN)^{T}$$?

d)    Take the transposes $$N^{T}$$ and $$M^{T}$$ and write out a few of their entries in the same form as in part (a).

e)    Multiply out $$N^{T}M^{T}$$ and write out a few of its entries in the same form as in part a. In terms of the entries of $$M$$ and the entries of $$N$$, what is the entry in row $$i$$ and column $$j$$ of $$N^{T}M^{T}$$?

f)    Show that the answers you got in parts (c) and (e) are the same.

3.
a)    Let $$A=\begin{pmatrix} 1 & 2 & 0\\ 3 & -1 & 4\\ \end{pmatrix}$$.  Find $$AA^{T}$$ and $$A^{T}A$$ and their traces.

b)    Let $$M$$ be any $$m\times n$$ matrix.  Show that $$M^{T}M$$ and $$MM^{T}$$ are symmetric. (Hint: use the result of the previous problem.)  What are their sizes? What is the relationship between their traces?

4.    Let $$x = \begin{pmatrix}x_{1} \\ \vdots \\ x_{n}\end{pmatrix}$$ and $$y = \begin{pmatrix}y_{1} \\ \vdots \\ y_{n}\end{pmatrix}$$ be column vectors.  Show that the dot product $$x \cdot y = x^{T}\ I\ y$$.

5.    Above, we showed that $$\textit{left}$$ multiplication by an $$r \times s$$ matrix $$N$$ was a linear transformation $$M^{s}_{k} \stackrel{N}{\longrightarrow} M^{r}_{k}$$.  Show that $$\textit{right}$$ multiplication by a $$k \times m$$ matrix $$R$$ is a linear transformation $$M^{s}_{k} \stackrel{R}{\longrightarrow} M^{s}_{m}$$.  In other words, show that right matrix multiplication obeys linearity.

6.    Let the $$V$$ be a vector space where $$B=(v_{1},v_{2})$$ is an ordered basis. Suppose
$$L:V\stackrel{\rm linear}{-\!\!-\!\!\!\longrightarrow} V$$
and $$L(v_{1})=v_{1}+v_{2} \, ,\quad L(v_{2})=2v_{1}+v_{2}\, .$$ Compute the matrix of $$L$$ in the basis $$B$$ and then compute the trace of this matrix. Suppose that $$ad-bc\neq 0$$ and consider now the new basis
$$B'=(av_{1}+b v_{2},cv_{1}+dv_{2})\, .$$
Compute the matrix of $$L$$ in the basis $$B'$$. Compute the trace of this matrix. What do you find? What do you conclude about the trace of a matrix? Does it make sense to talk about the trace of a linear transformation''?

7.    Explain what happens to a matrix when:

a)    You multiply it on the left by a diagonal matrix?
b)    You multiply it on the right by a diagonal matrix?

Give a few simple examples before you start explaining.

8.    Compute $$\exp (A)$$ for the following matrices:

a)    $$A = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \\ \end{pmatrix}$$
b)    $$A = \begin{pmatrix} 1 & \lambda \\ 0 & 1 \\ \end{pmatrix}$$
c)    $$A = \begin{pmatrix} 0 & \lambda \\ 0 & 0 \\ \end{pmatrix}$$

9.    Let $$M = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ \end{pmatrix}$$.  Divide $$M$$ into named blocks, with one block the $$4\times4$$ identity matrix, and then multiply blocks to compute $$M^{2}$$.

10.    A matrix $$A$$ is called $$\textit{anti-symmetric}$$ (or skew-symmetric matrix) if $$A^{T} = -A$$. Show that for every $$n \times n$$ matrix $$M$$, we can write $$M = A + S$$ where $$A$$ is an anti-symmetric matrix and $$S$$ is a symmetric matrix.

$$\textit{Hint: What kind of matrix is \(M + M^{T}$$? How about $$M - M^{T}$$?}\)

11.    An example of an operation which is not associative is the cross product.

a)    Give a simple example of three vectors from 3-space $$u,v,w$$ such that $$u\times (v\times w) \neq (u\times v)\times w$$.
b)    We saw in chapter 1 that the operator $$B=u\times$$ (cross product with a vector) is a linear operator. It can therefore be written as a matrix (given an ordered basis such as the standard basis). How is it that composing such linear operators is non-associative even though matrix multiplication is associative?