3.2. EXERCISES 87

Procedure 3.1.14 Suppose A is an n × n matrix and it is desired to solve the systemAx = y,y = (y1, · · · , yn)T for x = (x1, · · · , xn)T . Then Cramer’s rule says

xi =detAi

detA

where Ai is obtained from A by replacing the ith column of A with the column (y1, · · · , yn)T .

The following theorem is of fundamental importance and ties together many of the ideaspresented above. It is proved in the next section.

Theorem 3.1.15 Let A be an n× n matrix. Then the following are equivalent.

1. A is one to one.

2. A is onto.

3. det (A) ̸= 0.

3.2 Exercises

1. Find the determinants of the following matrices.

(a)

 1 2 3

3 2 2

0 9 8

 (The answer is 31.)

(b)

 4 3 2

1 7 8

3 −9 3

(The answer is 375.)

(c)

1 2 3 2

1 3 2 3

4 1 5 0

1 2 1 2

, (The answer is −2.)

2. If A−1 exists, what is the relationship between det (A) and det(A−1

). Explain your

answer.

3. Let A be an n × n matrix where n is odd. Suppose also that A is skew symmetric.This means AT = −A. Show that det(A) = 0.

4. Is it true that det (A+B) = det (A) + det (B)? If this is so, explain why it is so andif it is not so, give a counter example.

5. Let A be an r×r matrix and suppose there are r−1 rows (columns) such that all rows(columns) are linear combinations of these r − 1 rows (columns). Show det (A) = 0.

6. Show det (aA) = an det (A) where here A is an n× n matrix and a is a scalar.

7. Suppose A is an upper triangular matrix. Show that A−1 exists if and only if allelements of the main diagonal are non zero. Is it true that A−1 will also be uppertriangular? Explain. Is everything the same for lower triangular matrices?