Section 2.3 Matrix Multiplication

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Definition: Let $\overrightarrow{x}$ be a vector in ${\mathbb{R}}^p$ and $A$ be a $m\times n$ matrix and $B$ be a $n\times p$ matrix then $B\overrightarrow{x}$ is a vector in ${\mathbb{R}}^n$ and then $A(B\overrightarrow{x})$ is a vector in ${\mathbb{R}}^m$. We call $A(B\overrightarrow{x})$ the composition of mappings.

 

Notice: $A(B\overrightarrow{x})$ is only defined that the number of columns of $A$ is the number of rows of $B$.

 

 

Question: The composition $A(B\overrightarrow{x})$ is a matrix transformation that map from ${\mathbb{R}}^p$ to ${\mathbb{R}}^m$. What is the matrix of the transformation?

Let $A=\left[\begin{array}{ccc}\overrightarrow{a_1}& \cdots & \overrightarrow{a_n}\end{array}\right]$ and $B=\left[\begin{array}{ccc}\overrightarrow{b_1}& \cdots & \overrightarrow{b_p}\end{array}\right]$, and $\overrightarrow{x}=\left[\begin{array}{c}{x}_1\\ ⋮\\ x_p\end{array}\right]$

then $B\overrightarrow{x}=x_1\overrightarrow{b_1}+\cdots +x_p\overrightarrow{b_p}$.

We have

$$\begin{gathered} A(B\overrightarrow{x})=A(x_1\overrightarrow{b_1}+\cdots +x_p\overrightarrow{b_p}) \\ =x_{1}A\overrightarrow{b_1}+\cdots +x_{p}A\overrightarrow{b_p} \\ =\left[\begin{array}{c}A\overrightarrow{b_1}\cdots A\overrightarrow{b_p}\end{array}\right]\left[\begin{array}{c}{x}_1\\ ⋮\\ x_p\end{array}\right] \\ =(AB)\overrightarrow{x} \end{gathered}$$

 

It is naturally that we call the matrix $\left[\begin{array}{c}A\overrightarrow{b_1}\cdots A\overrightarrow{b_p}\end{array}\right]$ the product of the matrices $A$ and $B$ which is written $AB$.

Definition: Let $A$ be a $m\times n$ matrix and $B$ be a $n\times p$ matrix with column vectors $\overrightarrow{b_1},\cdots ,\overrightarrow{b_p}$, then the product of $A$ and $B$ which is written $AB$ is the $m\times p$ matrix $\left[\begin{array}{c}A\overrightarrow{b_1}\cdots A\overrightarrow{b_p}\end{array}\right]$.

 

Example 1: Find $AB$ and $BA$ if they are defined, where $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& -1& 4\end{array}\right]$ and $B=\left[\begin{array}{cccc}2& 0& -1& 3\\ 0& 4& 6& 2\\ 3& 1& 0& 1\end{array}\right]$.

 

 

Exercise 1: Find $AB$ and $BA$ if they are defined, where $A=\left[\begin{array}{ccc}1& -2& 0\\ 2& 1& 1\\ 3& -2& 0\\ 0& 1& 2\end{array}\right]$ and $B=\left[\begin{array}{cccc}2& 0& -1& 3\\ 0& 4& 6& 2\end{array}\right]$.

 

Definition: Given two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ in ${\mathbb{R}}^n$. The scalar product or the dot product or inner product of $\overrightarrow{u}$ and $\overrightarrow{v}$ is $(\overrightarrow{u}{)}^T\overrightarrow{v}$. The outer product of $\overrightarrow{u}$ and $\overrightarrow{v}$ is $(\overrightarrow{u}\overrightarrow{v}{)}^T$.

 

Example 2: Find the inner product and the outer product of $\overrightarrow{u}=\left[\begin{array}{c}2\\ 1\end{array}\right]$ and $\overrightarrow{v}=\left[\begin{array}{c}2\\ 1\end{array}\right]$.

 

 

Exercise 2: Find the inner product and the outer product of $\overrightarrow{u}=\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]$ and $\overrightarrow{v}=\left[\begin{array}{c}-1\\ 0\\ 3\end{array}\right]$.

 

Theorem: (Dot Product Rule) Let $A$ and $B$ be matrices of sizes $m\times n$ and $n\times p$, respectively. Then the $(i,j)$-entry of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

 

Example 3: Find $AB$ and $BA$ where $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$, and $B=\left[\begin{array}{cc}0& 2\\ 4& 5\end{array}\right]$.

What can you say about the relationship of $AB$ and $BA$?

 

 

Exercise 3: Find $AB$ and $AC$ where $A=\left[\begin{array}{cc}3& -6\\ -1& 2\end{array}\right]$, $B=\left[\begin{array}{cc}0& 2\\ 4& 5\end{array}\right]$, $C=\left[\begin{array}{cc}-3& -5\\ 2& 1\end{array}\right]$. What can you say about the relationship of $AB$ and $AC$? The relationship of $B$ and $C$?

 

Remark: 1. In general $AB\ne BA$.

 

2. When $AB=AC$, we cannot conclude that $B=C$ as in exercise 3.

 

3. When $AB=0$, we cannot conclude that $A$ or $B$ is 0.

 

Theorem: Let $A$ be a $m\times n$ matrix, $B$ and $C$ be matrices of the same size such that the products with $A$ are defined, and $r$ be a scalar. Then

a. $A(BC)=(AB)C$

b. $A(B+C)=AB+AC$

c. $(B+C)A=BA+CA$

d. $r(AB)=(rA)B=A(rB)$

e. $I_{m}A=A=A{I}_n$ where I_{m} is the $m\times m$ identity matrix and $I_n$ is the $n\times n$ identity matrix.

f. $(AB{)}^T=B^T{A}^T$

 

 

Example 4: Find $A^T$, $B^T$ and $(BA{)}^T$ where $A=\left[\begin{array}{ccc}1& 2& 3\\ -1& 3& 2\\ 0& -1& 1\end{array}\right]$, and $B=\left[\begin{array}{ccc}2& -1& 3\\ 1& 3& 2\end{array}\right]$.

 

 

Exercise 4: Find $A^T$, $B^T$ and $(BA{)}^T$ where $A=\left[\begin{array}{cc}2& 3\\ 0& 4\\ -1& 2\end{array}\right]$, and $B=\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]$.

 

Definition: Let $A$ be a $n\times n$ matrix then $A^k$ denotes the product of $k$ copies of $A$, i.e. $A^k=A...A$ for $k$ times

 

Example 5: Find $A^3$ where $A=\left[\begin{array}{ccc}1& 0& 3\\ -1& 3& 2\\ 0& -1& 1\end{array}\right]$.

 

 

Exercise 5: Find $B^4$ where $B=\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]$.

 

Group Work 1: Mark each statement True or False. Justify each answer.

a. If $A$ and $B$ are $2\times 2$ matrix with columns $\overrightarrow{a_1},\overrightarrow{a_2}$ and $\overrightarrow{b_1},\overrightarrow{b_2}$. Then the product $AB=\left[\begin{array}{cc}\overrightarrow{a_1}\overrightarrow{b_1}& \overrightarrow{a_2}\overrightarrow{b_2}\end{array}\right]$.

 

b. $A(B+C)=AB+BC$.

 

c. $A^T+B^T=(A+B{)}^T$.

 

d. The transpose of a product of matrices equals the product of their
transposes in the same order.

 

e. The first row of $AB$ is the first row of $A$ multiplied on the right by $B$.

 

f. $A$ is an $n\times n$ matrix , then $(A^2{)}^T=(A^T{)}^2$.

 

g. $(ABC{)}^T=C^T{A}^T{B}^T$.

 

GroupWork 2: Find $2\times 2$ nonzero matrices $A$ and $B$ such that

a. $A^2=0$

 

b. $A^2=A$

 

Group Work 3: If $A$ is any matrix, show that both $A{A}^T$ and $A^{T}A$ are symmetric.

 

Group Work 4: In each case either show the statement is true, or give an example showing that it is false.

a. If $A^2=I$, then $A=I$.

 

b. If $AJ=A$, then $J=I$.

 

c. If $A$ is symmetric, then $I+A$ is symmetric.

 

d. If $AB=AC$ and $A\ne 0$, then $B=C$.

 

e. If $A\ne 0$, then $A^2\ne 0$.

 

f. If $A$ has a row of zeros, so also does $BA$ for all $B$.

 

g. If $B$ has a column of zeros, so also does $AB$.

 

h. If $A$ has a row of zeros, so also does $AB$.