2.2. EXERCISES 53

15. Consider the following digraph.

1 2

3 4

Write the matrix associated with this digraph and find the number of ways to go from3 to 4 in three steps.

16. Show that if A−1 exists for an n×n matrix, then it is unique. That is, if BA = I andAB = I, then B = A−1.

17. Show (AB)−1

= B−1A−1.

18. Show that if A is an invertible n× n matrix, then so is AT and(AT)−1

=(A−1

)T.

19. Show that if A is an n×n invertible matrix and x is a n× 1 matrix such that Ax = bfor b an n× 1 matrix, then x = A−1b.

20. Give an example of a matrix A such that A2 = I and yet A ̸= I and A ̸= −I.

21. Give an example of matrices, A,B such that neither A nor B equals zero and yetAB = 0.

22. Write

x1 − x2 + 2x3

2x3 + x1

3x3

3x4 + 3x2 + x1

 in the form A

x1

x2

x3

x4

 where A is an appropriate matrix.

23. Give another example other than the one given in this section of two square matrices,A and B such that AB ̸= BA.

24. Suppose A and B are square matrices of the same size. Which of the following arecorrect?

(a) (A−B)2= A2 − 2AB +B2

(b) (AB)2= A2B2

(c) (A+B)2= A2 + 2AB +B2

(d) (A+B)2= A2 +AB +BA+B2

(e) A2B2 = A (AB)B

(f) (A+B)3= A3 + 3A2B + 3AB2 +B3

(g) (A+B) (A−B) = A2 −B2

(h) None of the above. They are all wrong.

(i) All of the above. They are all right.

25. Let A =

(−1 −1

3 3

). Find all 2× 2 matrices, B such that AB = 0.