Linear Algebra with Applications, 10th Edition

Section 1.4 Exercises

1. Explain why each of the following algebraic rules will not work, in general, when the real numbers a and b are replaced by $n\times n$ matrices A and B:

   1. $(a+b{)}^2=a^2+2ab+b^2$

   2. $(a+b)(a-b)=a^2-b^2$

2. Will the rules in Exercise 1 work if a is replaced by an $n\times n$ matrix A and b is replaced by the $n\times n$ identity matrix I?

3. Find nonzero $2\times 2$ matrices A and B such that $AB=O$.

4. Find nonzero matrices A, B, and C such that

   $$\begin{array}{ccc}AC=BC& \text{ and }& A\ne B\end{array}$$

5. The matrix

   $$A=\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]$$

   has the property that $A^2=O$. Is it possible for a nonzero symmetric $2\times 2$ matrix to have this property? Prove your answer.

6. Prove the associative law of multiplication for $2\times 2$ matrices; that is, let

   $$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right],B=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right],C=\left[\begin{array}{cc}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}\right]$$

   and show that

   $$\left(AB\right)C=A\left(BC\right)$$

7. Let