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

Remarks:

• Notice the absence of the “law” $$A B = B A\text{.}$$ Why?

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

### Exercises5.3.3Exercises

#### 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 $$A+B=B+A\text{,}$$

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

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

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

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

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

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

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

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

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

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

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

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, $$\left(A^{-1}\right)^{-1}=A\text{.}$$

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

#### 2.

Verify each of the Laws of Matrix Algebra using examples.

#### 3.

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

1. $$\displaystyle A B + A C$$

2. $$\displaystyle A^{-1}$$

3. $$\displaystyle A(B + C)$$

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

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

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

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

3. $$A(B+C)=A B+ B C\text{,}$$ which is given in part (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 $$B =\left( \begin{array}{cc} 3 & 5 \\ 2 & 4 \\ \end{array} \right)\text{.}$$ Compute the following as efficiently as possible by using any of the Laws of Matrix Algebra:

1. $$\displaystyle A B$$

2. $$\displaystyle A + B$$

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

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

5. $$\displaystyle A^2 + A B$$

#### 5.

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