Linear Algebra with Applications, 10th Edition

1.4

1. 1. Hint: In general

   $$BA+AB\ne 2AB\text{and}BA-AB\ne O$$

2. 3. Hint: There are many possible choices for A and B. Make up a nonzero matrix A whose first and second rows are the same and then see if you can come up with a matrix B such that $AB=O$.

3. 4. Hint: Make use of the result from Exercise3 .

4. 5. Hint: A $2\times 2$ symmetric matrix is one of the form

   $$\begin{array}{ccc}A& =& \left[\begin{array}{cc}a& b\\ b& c\end{array}\right]\end{array}$$

   If A is of this form and $A^2=O,$ what would that imply about the scalars a, b and c?

5. 7. $A=A^2=A^3=A^n$

6. 8. $A^{2n}=I,A^{2n+1}=A$

7. 10. Hint: If the given matrix is not necessarily symmetric, make up an example to show that the matrix need not be symmetric. If the matrix is necessarily symmetric, use the algebraic rules for transposes to show that the matrix must be symmetric.

8. 11. The hint for Exercise10 also applies to this ...