2.4E: Matrix Multiplication Exercises

Exercises for 1

solutions

2

Compute the following matrix products.

$\left[ \begin{array}{rr} 1 & 3 \\ 0 & -2 \end{array} \right] \left[ \begin{array}{rr} 2 & -1 \\ 0 & 1 \end{array} \right]$ $\left[ \begin{array}{rrr} 1 & -1 & 2 \\ 2 & 0 & 4 \end{array} \right] \left[ \begin{array}{rrr} 2 & 3 & 1 \\ 1 & 9 & 7 \\ -1 & 0 & 2 \end{array} \right]$ $\left[ \begin{array}{rrr} 5 & 0 & -7 \\ 1 & 5 & 9 \end{array} \right] \left[ \begin{array}{r} 3 \\ 1 \\ -1 \\ \end{array} \right]$ $\left[ \begin{array}{rrr} 1 & 3 & -3 \end{array} \right] \left[ \begin{array}{rr} 3 & 0 \\ -2 & 1 \\ 0 & 6 \end{array} \right]$ $\left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{rr} 3 & -2 \\ 5 & -7 \\ 9 & 7 \end{array} \right]$ $\left[ \begin{array}{rrr} 1 & -1 & 3 \end{array} \right] \left[ \begin{array}{r} 2 \\ 1 \\ -8 \end{array} \right]$ $\left[ \begin{array}{r} 2 \\ 1 \\ -7 \end{array} \right] \left[ \begin{array}{rrr} 1 & -1 & 3 \end{array} \right]$ $\left[ \begin{array}{rr} 3 & 1 \\ 5 & 2 \end{array} \right] \left[ \begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array} \right]$ $\left[ \begin{array}{rrr} 2 & 3 & 1 \\ 5 & 7 & 4 \end{array} \right] \left[ \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array} \right]$ $\left[ \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array} \right] \left[ \begin{array}{ccc} a^\prime & 0 & 0 \\ 0 & b^\prime & 0 \\ 0 & 0 & c^\prime \end{array} \right]$

2. $\left[ \begin{array}{rrr} -1 & -6 & -2 \\ 0 & 6 & 10 \end{array} \right]$
3. $\left[ \begin{array}{rr} -3 & -15 \end{array} \right]$
4. $\left[ -23 \right]$
5. $\left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]$
6. $\left[ \begin{array}{rrr} aa^\prime & 0 & 0 \\ 0 & bb^\prime & 0 \\ 0 & 0 & cc^\prime \end{array} \right]$

In each of the following cases, find all possible products $A^{2}$, $AB$, $AC$, and so on.

1. $A = \left[ \begin{array}{rrr} 1 & 2 & 3 \\ -1 & 0 & 0 \end{array} \right]$, $B = \left[ \begin{array}{rr} 1 & -2 \\ \frac{1}{2} & 3 \end{array} \right]$,
   $C = \left[ \begin{array}{rr} -1 & 0 \\ 2 & 5 \\ 0 & 3 \end{array} \right]$

2. $A = \left[ \begin{array}{rrr} 1 & 2 & 4 \\ 0 & 1 & -1 \end{array} \right]$, $B = \left[ \begin{array}{rr} -1 & 6 \\ 1 & 0 \end{array} \right]$,
   $C = \left[ \begin{array}{rr} 2 & 0 \\ -1 & 1 \\ 1 & 2 \end{array} \right]$

2. $BA = \left[ \begin{array}{rrr} -1 & 4 & -10 \\ 1 & 2 & 4 \end{array} \right]$, $B^{2} = \left[ \begin{array}{rr} 7 & -6 \\ -1 & 6 \end{array} \right]$, $CB = \left[ \begin{array}{rr} -2 & 12 \\ 2 & -6 \\ 1 & 6 \end{array} \right]$
   $AC = \left[ \begin{array}{rr} 4 & 10 \\ -2 & -1 \end{array} \right]$, $CA = \left[ \begin{array}{rrr} 2 & 4 & 8 \\ -1 & -1 & -5 \\ 1 & 4 & 2 \end{array} \right]$

Find $a$, $b$, $a_{1}$, and $b_{1}$ if:

1. $\left[ \begin{array}{cc} a & b \\ a_{1} & b_{1} \end{array} \right] \left[ \begin{array}{rr} 3 & -5 \\ -1 & 2 \end{array} \right] = \left[ \begin{array}{rr} 1 & -1 \\ 2 & 0 \end{array} \right]$
2. $\left[ \begin{array}{rr} 2 & 1 \\ -1 & 2 \end{array} \right] \left[ \begin{array}{cc} a & b \\ a_{1} & b_{1} \end{array} \right] = \left[ \begin{array}{rr} 7 & 2 \\ -1 & 4 \end{array} \right]$

2. $(a, b, a_{1}, b_{1}) = (3, 0, 1, 2)$

Verify that $A^{2} - A - 6I = 0$ if:

$\left[ \begin{array}{rr} 3 & -1 \\ 0 & -2 \end{array} \right]$ $\left[ \begin{array}{rr} 2 & 2 \\ 2 & -1 \end{array} \right]$

2. $A^{2} - A - 6I = \left[ \begin{array}{rr} 8 & 2 \\ 2 & 5 \end{array} \right] - \left[ \begin{array}{rr} 2 & 2 \\ 2 & -1 \end{array} \right] - \left[ \begin{array}{rr} 6 & 0 \\ 0 & 6 \end{array} \right] = \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]$

[ex:ex2_3_5]

Given $A = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array} \right]$, $B = \left[ \begin{array}{rrr} 1 & 0 & -2 \\ 3 & 1 & 0 \end{array} \right]$,
$C = \left[ \begin{array}{rr} 1 & 0 \\ 2 & 1 \\ 5 & 8 \end{array} \right]$, and $D = \left[ \begin{array}{rrr} 3 & -1 & 2 \\ 1 & 0 & 5 \end{array} \right]$, verify the following facts from Theorem [thm:003469].

$A(B - D) = AB - AD$ $A(BC) = (AB)C$ $(CD)^{T} = D^{T}C^{T}$

2. $A(BC) = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{rr} -9 & -16 \\ 5 & 1 \end{array} \right] = \left[ \begin{array}{rr} -14 & -17 \\ 5 & 1 \end{array} \right] = \left[ \begin{array}{rrr} -2 & -1 & -2 \\ 3 & 1 & 0 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ 2 & 1 \\ 5 & 8 \end{array} \right] = (AB)C$

Let $A$ be a $2 \times 2$ matrix.

1. If $A$ commutes with $\left[ \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right]$, show that
   $A = \left[ \begin{array}{rr} a & b \\ 0 & a \end{array} \right]$ for some $a$ and $b$.

2. If $A$ commutes with $\left[ \begin{array}{rr} 0 & 0 \\ 1 & 0 \end{array} \right]$, show that
   $A = \left[ \begin{array}{rr} a & 0 \\ c & a \end{array} \right]$ for some $a$ and $c$.

3. Show that $A$ commutes with every $2 \times 2$ matrix if and only if $A = \left[ \begin{array}{rr} a & 0 \\ 0 & a \end{array} \right]$ for some $a$.

2. If $A = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]$ and $E = \left[ \begin{array}{rr} 0 & 0 \\ 1 & 0 \end{array} \right]$, compare entries an $AE$ and $EA$.

1. If $A^{2}$ can be formed, what can be said about the size of $A$?
2. If $AB$ and $BA$ can both be formed, describe the sizes of $A$ and $B$.
3. If $ABC$ can be formed, $A$ is $3 \times 3$, and $C$ is $5 \times 5$, what size is $B$?

2. $m \times n$ and $n \times m$ for some $m$ and $n$

1. Find two $2 \times 2$ matrices $A$ such that $A^{2} = 0$.
2. Find three $2 \times 2$ matrices $A$ such that (i) $A^{2} = I$; (ii) $A^{2} = A$.
3. Find $2 \times 2$ matrices $A$ and $B$ such that $AB = 0$ but $BA \neq 0$.

2. 1. $\left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]$, $\left[ \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right]$, $\left[ \begin{array}{rr} 1 & 1 \\ 0 & -1 \end{array} \right]$
   2. $\left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right]$, $\left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]$, $\left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right]$

Write $P = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]$, and let $A$ be $3 \times n$ and $B$ be $m \times 3$.

1. Describe $PA$ in terms of the rows of $A$.
2. Describe $BP$ in terms of the columns of $B$.

Let $A$, $B$, and $C$ be as in Exercise [ex:ex2_3_5]. Find the $(3, 1)$-entry of $CAB$ using exactly six numerical multiplications.

Compute $AB$, using the indicated block partitioning.

$$A = \left[ \begin{array}{rr|rr} 2 & -1 & 3 & 1 \\ 1 & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \quad B = \left[ \begin{array}{rr|r} 1 & 2 & 0 \\ -1 & 0 & 0 \\ \hline 0 & 5 & 1 \\ 1 & -1 & 0 \end{array} \right] \nonumber$$

In each case give formulas for all powers $A, A^{2}, A^{3}, \dots$ of $A$ using the block decomposition indicated.

1. $A = \left[ \begin{array}{r|rr} 1 & 0 & 0 \\ \hline 1 & 1 & -1\\ 1 & -1 & 1 \end{array} \right]$
2. $A = \left[ \begin{array}{rr|rr} 1 & -1 & 2 & -1 \\ 0 & 1 & 0 & 0 \\ \hline 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]$

2. $A^{2k} = \left[ \begin{array}{rc|rr} 1 & -2k & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \hline 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$ for $k = 0, 1, 2, \dots$,
   $A^{2k + 1} = A^{2k}A = \left[ \begin{array}{rc|rr} 1 & -(2k + 1) & 2 & -1 \\ 0 & 1 & 0 & 0 \\ \hline 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]$ for $k = 0, 1, 2, \dots$

Compute the following using block multiplication (all blocks are $k \times k$).

$\left[ \begin{array}{rr} I & X \\ -Y & I \end{array} \right] \left[ \begin{array}{rr} I & 0 \\ Y & I \end{array} \right]$ $\left[ \begin{array}{rr} I & X \\ 0 & I \end{array} \right] \left[ \begin{array}{rr} I & -X \\ 0 & I \end{array} \right]$ $\left[ \begin{array}{cc} I & X \end{array} \right] \left[ \begin{array}{cc} I & X \end{array} \right]^{T}$ $\left[ \begin{array}{cc} I & X^{T} \end{array} \right] \left[ \begin{array}{cc} -X & I \end{array} \right]^{T}$ $\left[ \begin{array}{cc} I & X \\ 0 & -I \end{array} \right]^{n}$ any $n \geq 1$ $\left[ \begin{array}{cc} 0 & X \\ I & 0 \end{array} \right]^{n}$ any $n \geq 1$

2. $\left[ \begin{array}{cc} I & 0 \\ 0 & I \end{array} \right] = I_{2k}$
3. $0_{k}$
4. $\left[ \begin{array}{cc} X^{m} & 0 \\ 0 & X^{m} \end{array} \right]$ if $n = 2m$; $\left[ \begin{array}{cc} 0 & X^{m + 1} \\ X^{m} & 0 \end{array} \right]$ if $n = 2m + 1$

Let $A$ denote an $m \times n$ matrix.

1. If $AX = 0$ for every $n \times 1$ matrix $X$, show that $A = 0$.
2. If $YA = 0$ for every $1 \times m$ matrix $Y$, show that $A = 0$.

2. If $Y$ is row $i$ of the identity matrix $I$, then $YA$ is row $i$ of $IA = A$.

1. If $U = \left[ \begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array} \right]$, and $AU = 0$, show that $A = 0$.
2. Let $U$ be such that $AU = 0$ implies that $A = 0$. If $PU = QU$, show that $P = Q$.

Simplify the following expressions where $A$, $B$, and $C$ represent matrices.

1. $A(3B - C) + (A - 2B)C + 2B(C + 2A)$
2. $A(B + C - D) + B(C - A + D) - (A + B)C \\ + (A - B)D$
3. $AB(BC - CB) + (CA - AB)BC + CA(A - B)C$
4. $(A - B)(C - A) + (C - B)(A - C) + (C - A)^{2}$

2. $AB - BA$
3. $0$

If $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]$ where $a \neq 0$, show that $A$ factors in the form $A = \left[ \begin{array}{cc} 1 & 0 \\ x & 1 \end{array} \right] \left[ \begin{array}{cc} y & z \\ 0 & w \end{array} \right]$.

If $A$ and $B$ commute with $C$, show that the same is true of:

$A + B$ $kA$, $k$ any scalar

2. $(kA)C = k(AC) = k(CA) = C(kA)$

If $A$ is any matrix, show that both $AA^{T}$ and $A^{T}A$ are symmetric.

If $A$ and $B$ are symmetric, show that $AB$ is symmetric if and only if $AB = BA$.

We have $A^{T} = A$ and $B^{T} = B$, so $(AB)^{T} = B^{T}A^{T} = BA$. Hence $AB$ is symmetric if and only if $AB = BA$.

If $A$ is a $2 \times 2$ matrix, show that $A^{T}A = AA^{T}$ if and only if $A$ is symmetric or $A = \left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$ for some $a$ and $b$.

1. Find all symmetric $2 \times 2$ matrices $A$ such that $A^{2} = 0$.
2. Repeat (a) if $A$ is $3 \times 3$.
3. Repeat (a) if $A$ is $n \times n$.

2. $A = 0$

Show that there exist no $2 \times 2$ matrices $A$ and $B$ such that $AB - BA = I$. [Hint: Examine the $(1, 1)$- and $(2, 2)$-entries.]

Let $B$ be an $n \times n$ matrix. Suppose $AB = 0$ for some nonzero $m \times n$ matrix $A$. Show that no $n \times n$ matrix $C$ exists such that $BC = I$.

If $BC = I$, then $AB = 0$ gives $0 = 0C = (AB)C = A(BC) = AI = A$, contrary to the assumption that $A \neq 0$.

An autoparts manufacturer makes fenders, doors, and hoods. Each requires assembly and packaging carried out at factories: Plant 1, Plant 2, and Plant 3. Matrix $A$ below gives the number of hours for assembly and packaging, and matrix $B$ gives the hourly rates at the three plants. Explain the meaning of the $(3, 2)$-entry in the matrix $AB$. Which plant is the most economical to operate? Give reasons.

lccll & Assembly & Packaging & &
$\begin{array}{l} \mbox{Fenders} \\ \mbox{Doors} \\ \mbox{Hoods} \end{array}$ & & $=$ & $A$

For the directed graph below, find the adjacency matrix $A$, compute $A^{3}$, and determine the number of paths of length $3$ from $v_{1}$ to $v_{4}$ and from $v_{2}$ to $v_{3}$.

$3$ paths $v_{1} \rightarrow v_{4}$, $0$ paths $v_{2} \rightarrow v_{3}$

In each case either show the statement is true, or give an example showing that it is false.

1. If $A^{2} = I$, then $A = I$.
2. If $AJ = A$, then $J = I$.
3. If $A$ is square, then $(A^{T})^{3} = (A^{3})^{T}$.
4. If $A$ is symmetric, then $I + A$ is symmetric.
5. If $AB = AC$ and $A \neq 0$, then $B = C$.
6. If $A \neq 0$, then $A^{2} \neq 0$.
7. If $A$ has a row of zeros, so also does $BA$ for all $B$.
8. If $A$ commutes with $A + B$, then $A$ commutes with $B$.
9. If $B$ has a column of zeros, so also does $AB$.
10. If $AB$ has a column of zeros, so also does $B$.
11. If $A$ has a row of zeros, so also does $AB$.
12. If $AB$ has a row of zeros, so also does $A$.

2. False. If $A = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right] = J$, then $AJ = A$ but $J \neq I$.
3. True. Since $A^{T} = A$, we have $(I + AT = I^{T} + A^{T} = I + A$.
4. False. If $A = \left[ \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right]$, then $A \neq 0$ but $A^{2} = 0$.
5. True. We have $A(A + B) = (A + B)A$; that is, $A^{2} + AB = A^{2} + BA$. Subtracting $A^{2}$ gives $AB = BA$.
6. False. $A = \left[ \begin{array}{rr} 1 & -2 \\ 2 & 4 \end{array} \right]$, $B = \left[ \begin{array}{rr} 2 & 4 \\ 1 & 2 \end{array} \right]$
7. False. See (j).

1. If $A$ and $B$ are $2 \times 2$ matrices whose rows sum to $1$, show that the rows of $AB$ also sum to $1$.
2. Repeat part (a) for the case where $A$ and $B$ are $n \times n$.

2. If $A = \left[ a_{ij} \right]$ and $B = \left[ b_{ij} \right]$ and $\sum_{j}a_{ij} = 1 = \sum_{j}b_{ij}$, then the $(i, j)$-entry of $AB$ is $c_{ij} = \sum_{k}a_{ik}b_{kj}$, whence $\sum_{j}c_{ij} = \sum_{j}\sum_{k}a_{ik}b_{kj} = \sum_{k}a_{ik}(\sum_{j}b_{kj}) = \sum_{k}a_{ik} = 1$. Alternatively: If $\mathbf{e} = (1, 1, \dots, 1)$, then the rows of $A$ sum to $1$ if and only if $A\mathbf{e} = \mathbf{e}$. If also $B\mathbf{e} = \mathbf{e}$ then $(AB)\mathbf{e} = A(B\mathbf{e}) = A\mathbf{e} = \mathbf{e}$.

Let $A$ and $B$ be $n \times n$ matrices for which the systems of equations $A\mathbf{x} = \mathbf{0}$ and $B\mathbf{x} = \mathbf{0}$ each have only the trivial solution $\mathbf{x} = \mathbf{0}$. Show that the system $(AB)\mathbf{x} = \mathbf{0}$ has only the trivial solution.

The trace of a square matrix $A$, denoted $tr \;A$, is the sum of the elements on the main diagonal of $A$. Show that, if $A$ and $B$ are $n \times n$ matrices:

$\func{tr}(A + B) = tr \;A + tr \;B$. $\func{tr}(kA) = k \func{tr}(A)$ for any number $k$. $\func{tr}(A^{T}) = \func{tr}(A)$. $\func{tr}(AB) = \func{tr}(BA)$. $\func{tr}(AA^{T})$ is the sum of the squares of all entries of $A$.

2. If $A = \left[ a_{ij} \right]$, then $\func{tr}(kA) = \func{tr}\left[ ka_{ij} \right] = \sum_{i=1}^{n} ka_{ii} = k\sum_{i=1}^{n} a_{ii} = k \func{tr}(A)$.
3. Write $A^{T} = \left[ a_{ij}^\prime \right]$, where $a_{ij}^\prime = a_{ji}$. Then $AA^{T} = \left( \sum_{k=1}^{n} a_{ik}a_{kj}^\prime \right)$, so $\func{tr}(AA^{T}) = \sum_{i=1}^{n} \left[ \sum_{k=1}^{n} a_{ik}a_{ki}^\prime \right] = \sum_{i=1}^{n} \sum_{k=1}^{n} a_{ik}^{2}$.

Show that $AB - BA = I$ is impossible.

[Hint: See the preceding exercise.]

A square matrix $P$ is called an idempotent if $P^{2} = P$. Show that:

1. $0$ and $I$ are idempotents.
2. $\left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right]$, $\left[ \begin{array}{rr} 1 & 0 \\ 1 & 0 \end{array} \right]$, and $\frac{1}{2} \left[ \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right]$, are idempotents.
3. If $P$ is an idempotent, so is $I - P$. Show further that $P(I - P) = 0$.
4. If $P$ is an idempotent, so is $P^{T}$.
5. If $P$ is an idempotent, so is $Q = P + AP - PAP$ for any square matrix $A$ (of the same size as $P$).
6. If $A$ is $n \times m$ and $B$ is $m \times n$, and if $AB = I_{n}$, then $BA$ is an idempotent.

5. Observe that $PQ = P^{2} + PAP - P^{2}AP = P$, so $Q^{2} = PQ + APQ - PAPQ = P + AP - PAP = Q$.

Let $A$ and $B$ be $n \times n$ diagonal matrices (all entries off the main diagonal are zero).

1. Show that $AB$ is diagonal and $AB = BA$.
2. Formulate a rule for calculating $XA$ if $X$ is $m \times n$.
3. Formulate a rule for calculating $AY$ if $Y$ is $n \times k$.

If $A$ and $B$ are $n \times n$ matrices, show that:

1. $AB = BA$ if and only if

   $$(A + B)^{2} = A^{2} + 2AB + B^{2} \nonumber$$

2. $AB = BA$ if and only if

   $$(A + B)(A - B) = (A - B)(A + B) \nonumber$$

2. $(A + B)(A - B) = A^{2} - AB + BA - B^{2}$, and $(A - B)(A + B) = A^{2} + AB - BA - B^{2}$. These are equal if and only if $-AB + BA = AB - BA$; that is, $2BA = 2AB$; that is, $BA = AB$.

In Theorem [thm:003584], prove

part 3; part 5.

2. $(A + B)(A - B) = A^{2} - AB + BA - B^{2}$ and $(A - B)(A + B) = A^{2} - BA + AB - B^{2}$. These are equal if and only if $-AB + BA = -BA + AB$, that is $2AB = 2BA$, that is $AB = BA$.