$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

## Section5.3Laws of Matrix Algebra

### Subsection5.3.1The Laws

The following is a summary of the basic laws of matrix operations. Assume that the indicated operations are defined; that is, that the orders of the matrices $A\text{,}$ $B$ and $C$ are such that the operations make sense.

### Subsection5.3.2Commentary

If we wished to write out each of the above laws more completely, we would specify the orders of the matrices. For example, Law 10 should read:

Let $A\text{,}$ $B\text{,}$ and $C$ be $m\times n$, $n\times p$, and $n\times p$ matrices, respectively, then $A(B + C) = A B + A C$

Remarks:

• Notice the absence of the “law” $A B = B A$. Why?

• Is it really necessary to have both a right (No. 11) and a left (No. 10) distributive law? Why?

### SubsectionExercises

###### 1

Rewrite the above laws specifying as in Example 5.3.2 the orders of the matrices.

1. Let $A$ and $B$ be $m$ by $n$ matrices. Then $A+B=B+A$,

2. Let $A$, $B$, and $C$ be $m$ by $n$ matrices. Then $A+(B+C)=(A+B)+C$.

3. Let $A$ and $B$ be $m$ by $n$ matrices, and let $c\in \mathbb{R}$. Then $c(A+B)=cA+cB$,

4. Let $A$ be an $m$ by $n$ matrix, and let $c_1,c_2\in \mathbb{R}$. Then $\left(c_1+c_2\right)A=c_1A+c_2A$.

5. Let $A$ be an $m$ by $n$ matrix, and let $c_1,c_2\in \mathbb{R}$. Then $c_1\left(c_2A\right)=\left(c_1c_2\right)A$

6. Let $\pmb{0}$ be the zero matrix, of size $m \textrm{ by } n$, and let $A$ be a matrix of size $n \textrm{ by } r$. Then $\pmb{0}A=\pmb{0}=\textrm{ the } m \textrm{ by } r \textrm{ zero matrix}$.

7. Let $A$ be an $m \textrm{ by } n$ matrix, and $0 = \textrm{ the number zero}$. Then $0A=0=\textrm{ the } m \textrm{ by } n \textrm{ zero matrix}$.

8. Let $A$ be an $m \textrm{ by } n$ matrix, and let $\pmb{0}$ be the $m \textrm{ by } n$ zero matrix. Then $A+\pmb{0}=A$.

9. Let $A$ be an $m \textrm{ by } n$ matrix. Then $A+(- 1)A=\pmb{0}$, where $\pmb{0}$ is the $m \textrm{ by } n$ zero matrix.

10. Let $A$, $B$, and $C$ be $m \textrm{ by } n$, $n \textrm{ by } r$, and $n \textrm{ by } r$ matrices respectively. Then $A(B+C)=AB+AC$.

11. Let $A$, $B$, and $C$ be $m \textrm{ by } n$, $r \textrm{ by } m$, and $r \textrm{ by } m$ matrices respectively. Then $(B+C)A=BA+CA$.

12. Let $A$, $B$, and $C$ be $m \textrm{ by } n$, $n \textrm{ by } r$, and $r \textrm{ by } p$ matrices respectively. Then $A(BC)=(AB)C$.

13. Let $A$ be an $m \textrm{ by } n$ matrix, $I_m$ the $m \textrm{ by } m$ identity matrix, and $I_n$ the $n \textrm{ by } n$ identity matrix. Then $I_mA=AI_n=A$

14. Let $A$ be an $n \textrm{ by } n$ matrix. Then if $A^{-1}$ exists, $\left(A^{-1}\right)^{-1}=A$.

15. Let $A$ and $B$ be $n \textrm{ by } n$ matrices. Then if $A^{-1}$ and $B^{-1}$ exist, $(AB)^{-1}=B^{-1}A^{-1}$.

###### 2

Verify each of the Laws of Matrix Algebra using examples.

###### 3

Let $A = \left( \begin{array}{cc} 1 & 2 \\ 0 & -1 \\ \end{array} \right)$, $B= \left( \begin{array}{ccc} 3 & 7 & 6 \\ 2 & -1 & 5 \\ \end{array} \right)$, and $C= \left( \begin{array}{ccc} 0 & -2 & 4 \\ 7 & 1 & 1 \\ \end{array} \right)$. Compute the following as efficiently as possible by using any of the Laws of Matrix Algebra:

1. $A B + A C$

2. $A^{-1}$

3. $A(B + C)$

4. $\left(A^2\right)^{-1}$

5. $(C + B)^{-1}A^{-1}$

1. $AB+AC=\left( \begin{array}{ccc} 21 & 5 & 22 \\ -9 & 0 & -6 \\ \end{array} \right)$

2. $A(B+C)=A B+ B C$

3. $A^{-1}=\left( \begin{array}{cc} 1 & 2 \\ 0 & -1 \\ \end{array} \right)=A$

4. $\left(A^2\right)^{-1}=(AA)^{-1}=(A^{-1}A)=I^{-1}=I \quad$ by part c

###### 4

Let $A =\left( \begin{array}{cc} 7 & 4 \\ 2 & 1 \\ \end{array} \right)$ and $B =\left( \begin{array}{cc} 3 & 5 \\ 2 & 4 \\ \end{array} \right)$. Compute the following as efficiently as possible by using any of the Laws of Matrix Algebra:

1. $A B$

2. $A + B$

3. $A^2 + A B + B A + B ^2$

4. $B^{-1}A^{-1}$

5. $A^2 + A B$

###### 5

Let $A$ and $B$ be $n\times n$ matrices of real numbers. Is $A^2-B^2= (A-B)(A+B)$? Explain.