1. If a computer takes 3 seconds to invert a 60 x 60 matrix, estimate how long will it take to invert a matrix of order 300?
   Answer:   About 31.25 minutes

2. Suppose A is square. Answer True or False:  If A has a zero in its diagonal, then A is singular.

3. Suppose we've used Gauss elimination to transform A into upper triangular form U.
   Answer True or False:  If U has a zero in its diagonal, then A is singular.

4. Answer True or False:  If A is singular, then using Gauss elimination to solve system Ax = b will result in dividing by 0.

5. Using Gauss elimination to solve system Ax = b will produce large truncation error when WHAT is true about A?

6. Using the inverse method is a
   1. very efficient
   2. very inefficient
   way to solve a system Ax = b.

7. Given matrix A :

   A   = 4 –19 –7 33

   1. Find A^–1 using the method covered in class.
   2. Then show that AA^–1 = I and that A^–1A = I .
   3. What is det(A) ?
   4. What is det(A^–1) ?
   Fall 2024 Students: Do Not work the problems below.

8. Use the inverse method to solve the system

   4 –19 –7 33

   x1 x2

   =

   12 –5

   Use Gauss elimination or Maple to check your answer.

9. See this Maple example for inverting a matrix.

   You may then use Maple to check your answers to problems 1 and 2.

10. Use Maple (as above) to obtain the inverse of coefficient matrix A:

    		1	–1	0
    A	=	1	0	–1
    		–6	2	3

    Now use the inverse method to solve system

    1 –1 0 1 0 –1 –6 2 3

    x1 x2 x3

    =

    –2 4 3

    
    

    ---

    Do the following using an appropriate theorem covered in class (to be covered Thursday).

11. Without finding the inverse of A, calculate det(A^-1) for A.

    A   = 8     6 4     9

12. Prove the theorem that if A is invertible, then its inverse is unique.
    Hint: Suppose B and C are both inverses of A. Show that B = C.
    Warning: don't refer to either B or C as A^–1. By doing so you are unknowingly assuming what you are trying to prove.

13. If det(A) = 100,000 and A has order 12, what is det(¾A)?    (the number is three-fourths)
    Answer:   3,167.6352

14. If  | A | = 12,345  and A has order 8, what is | 4A^–1|  ?   
    Answer:   5.3087

15. Suppose both A and B are order n and invertible. Answer the following with True or False:
    1.   A^–1  =  1÷A
    2.   (A^–1)^–1  =  A
    3.   det(A)   =   det(A^–1)
    4.   det(A + B)   =   det(A) + det(B) 
    5.   (A + B)^–1   =   A^–1 + B^–1
    6.   (AB)^–1   =   A^–1 B^–1
    7.   (A + B)^T   =   A^T + B^T
    8.   | A^T |   =   | A |