1. 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) ?

2. Use the inverse method to solve the system

   4 –19 –7 33

   x1 x2

   =

   12 –5

   Use Gauss elimination, LU decomposition, or Maple to check your answer.

3. See this Maple example for inverting a matrix.

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

   
   

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   Work the problems below after Monday's lecture.

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

   A   = 8     6 4     9

5. If a computer takes 30 seconds to invert a 60 x 60 matrix, estimate how long will it take to invert a matrix of order 300?
   Answer:   About 5.2 hours

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

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

8. Answer true or false:   If a square matrix A has a zero in its diagonal, then A is automatically singular.