Mechanics of Matrix Multiplication

Predict the result of each product with a permutation matrix and then check by multiplying it out.

• (a)  $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$     (b)  $\begin{pmatrix}1&2\\ 3&4\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$     (c)  $\begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}\begin{pmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{pmatrix}$

Predict the result of each multiplication by an elementary reduction matrix, and then check by multiplying it out.

• (a)  $\begin{pmatrix}3&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$     (b)  $\begin{pmatrix}1&0\\ 0&2\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$     (c)  $\begin{pmatrix}1&0\\ -2&1\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$     (d)  $\begin{pmatrix}1&2\\ 3&4\end{pmatrix}\begin{pmatrix}1&-1\\ 0&1\end{pmatrix}$     (e)  $\begin{pmatrix}1&2\\ 3&4\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$

Predict the result of each multiplication by a diagonal matrix, and then check by multiplying it out.

• (a)  $\begin{pmatrix}-3&0\\ 0&0\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$     (b)  $\begin{pmatrix}4&0\\ 0&2\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$

Produce each.

• a $3\!\times\!3$ matrix that, acting from the left, swaps rows one and two

• a $2\!\times\!2$ matrix that, acting from the right, swaps column one and two

Show how to use matrix multiplication to bring this matrix to echelon form.

Find the product of this matrix with its transpose.

This table gives the number of hours of each type done by each worker, and the associated pay rates. Use matrices to compute the wages due.

Remark. This illustrates that in practice we often want to compute linear combinations of rows and columns in a context where we really aren’t interested in any associated linear maps.

Express this nonsingular matrix as a product of elementary reduction matrices.

Express

as the product of two elementary reduction matrices.

Prove that the diagonal matrices form a subspace of $\mathcal{M}_{n\!\times\!n}$. What is its dimension?

Does the identity matrix represent the identity map if the bases are unequal?

Show that every multiple of the identity commutes with every square matrix. Are there other matrices that commute with all square matrices?

Prove or disprove: nonsingular matrices commute.

Show that the product of a permutation matrix and its transpose is an identity matrix.

Show that if the first and second rows of $G$ are equal then so are the first and second rows of $GH$. Generalize.

Describe the product of two diagonal matrices.

Show that if $G$ has a row of zeros then $GH$ (if defined) has a row of zeros. Does that work for columns?

Show that the set of unit matrices forms a basis for $\mathcal{M}_{n\!\times\!m}$.

Find the formula for the $n$-th power of this matrix.

The trace of a square matrix is the sum of the entries on its diagonal (its significance appears in Chapter Five). Show that $\operatorname{Tr}(GH)=\operatorname{Tr}(HG)$.

A square matrix is upper triangular if its only nonzero entries lie above, or on, the diagonal. Show that the product of two upper triangular matrices is upper triangular. Does this hold for lower triangular also?

A square matrix is a Markov matrix if each entry is between zero and one and the sum along each row is one. Prove that a product of Markov matrices is Markov.

Give an example of two matrices of the same rank and size with squares of differing rank.

Matrix multiplication is performed often on computers. Researchers trying to understand its performance, and improve on it, count the number of operations that it takes.

• Definition 2.3 gives $p_{i,j}=g_{i,1}h_{1,j}+g_{i,2}h_{2,j}+\dots+g_{i,r}h_{r,j}$. How many real number multiplications are in that expression? Using it, how many do we need for the product of a $m\!\times\!r$ matrix and a $r\!\times\!n$ matrix?

• Matrix multiplication is associative, so in computing $H_{1}H_{2}H_{3}H_{4}$ we can ex…

If $A$ and $B$ are square matrices of the same size such that $ABAB=0$, does it follow that $BABA=0$?

Demonstrate these four assertions to get an alternate proof that column rank equals row rank.

• $\vec{y}\cdot\vec{y}=0$ iff $\vec{y}=\vec{0}$.

• $A\vec{x}=\vec{0}$ iff ${{A}^{\mathsf{T}}}A\vec{x}=\vec{0}$.

• $\dim(\mathscr{R}(A))=\dim(\mathscr{R}({{A}^{\mathsf{T}}}A))$.

• $\text{col rank}(A)=\text{col rank}({{A}^{\mathsf{T}}})=\text{row rank}(A)$.

Prove (where $A$ is an $n\!\times\!n$ matrix and so defines a transformation of any $n$-dimensional space $V$ with respect to $B,B$ where $B$ is a basis) that $\dim(\mathscr{R}(A)\cap\mathscr{N}(A))=\dim(\mathscr{R}(A))-\dim(\mathscr{R}(A^{2}))$. Conclude

• $\mathscr{N}(A)\subset\mathscr{R}(A)$ iff $\dim(\mathscr{N}(A))=\dim(\mathscr{R}(A))-\dim(\mathscr{R}(A^{2}))$;

• $\mathscr{R}(A)\subseteq\mathscr{N}(A)$ iff $A^{2}=0$;

• $\mathscr{R}(A)=\mathscr{N}(A)$ iff $A^{2}=0$ and $\dim(\mathscr{N}(A))=\dim(\mathscr{R}(A))$ ;

• $\dim(\mathscr{R}(A)\cap\mathscr{N}(A))=0$ iff $\dim(\mathscr{R}(A))=\dim(\mathscr{R}(A^{2}))$ ;

• (Requires the Direct Sum subsection, which is optional.) $V=\mathscr{R}(A)\oplus\mathscr{N}(A)$ iff $\dim(\mathscr{R}(A))=\dim(\mathscr{R}(A^{2}))$.