Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Determine whether the matrices are multiplicative inverses.

7. [ 3 2 4 3 ] , [ 3 − 2 − 4 3 ] . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 2 , row2 column 1 , 4 , column 2 3 , end matrix . comma . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 negative 2 , row2 column 1 , negative 4 , column 2 3 , end matrix
8. [ − 3 7 − 2 5 ] , [ − 5 7 − 2 3 ] . matrix with 2 rows and 2 columns , row1 column 1 , negative 3 , column 2 7 , row2 column 1 , negative 2 , column 2 5 , end matrix . comma . matrix with 2 rows and 2 columns , row1 column 1 , negative 5 , column 2 7 , row2 column 1 , negative 2 , column 2 3 , end matrix
9. [ 1 5 − 1 10 0 1 4 ] , [ 5 2 0 4 ] . matrix with 2 rows and 2 columns , row1 column 1 , 1 fifth , column 2 negative , 1 tenth , row2 column 1 , 0 , column 2 1 fourth , end matrix . comma . matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 2 , row2 column 1 , 0 , column 2 4 , end matrix
10. [ 1 2 − 1 − 1.5 − 3 1.75 0 − 1 0.5 ] , [ 1 0 2 3 2 − 1 6 4 0 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 2 , column 3 negative 1 , row2 column 1 , negative 1.5 , column 2 negative 3 , column 3 1.75 , row3 column 1 , 0 , column 2 negative 1 , column 3 0.5 , end matrix . comma . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 0 , column 3 2 , row2 column 1 , 3 , column 2 2 , column 3 negative 1 , row3 column 1 , 6 , column 2 4 , column 3 0 , end matrix
11. [ 2 2 2 − 2 2 − 2 − 2 − 2 − 2 ] , [ 2 2 2 − 2 2 − 2 − 2 − 2 − 2 ] . matrix with 3 rows and 3 columns , row1 column 1 , 2 , column 2 2 , column 3 2 , row2 column 1 , negative 2 , column 2 2 , column 3 negative 2 , row3 column 1 , negative 2 , column 2 negative 2 , column 3 negative 2 , end matrix . comma . matrix with 3 rows and 3 columns , row1 column 1 , 2 , column 2 2 , column 3 2 , row2 column 1 , negative 2 , column 2 2 , column 3 negative 2 , row3 column 1 , negative 2 , column 2 negative 2 , column 3 negative 2 , end matrix

See Problems 2, 3, and 4.

Evaluate the determinant of each matrix.

12. [ 7 2 0 − 3 ] . matrix with 2 rows and 2 columns , row1 column 1 , 7 , column 2 2 , row2 column 1 , 0 , column 2 negative 3 , end matrix
13. [ 6 2 − 6 − 2 ] . matrix with 2 rows and 2 columns , row1 column 1 , 6 , column 2 2 , row2 column 1 , negative 6 , column 2 negative 2 , end matrix
14. [ 0 0.5 1.5 2 ] . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 0.5 , row2 column 1 , 1.5 , column 2 2 , end matrix
15. [ 1 2 2 3 3 5 1 4 ] . matrix with 2 rows and 2 columns , row1 column 1 , 1 half , column 2 2 thirds , row2 column 1 , 3 fifths , column 2 1 fourth , end matrix
16. [ − 1 3 5 2 ] . matrix with 2 rows and 2 columns , row1 column 1 , negative 1 , column 2 3 , row2 column 1 , 5 , column 2 2 , end matrix
17. [ 5 3 − 2 1 ] . matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 3 , row2 column 1 , negative 2 , column 2 1 , end matrix
18. [ 2 − 1 5 − 4 ] . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 1 , row2 column 1 , 5 , column 2 negative 4 , end matrix
19. [ − 4 3 2 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , negative 4 , column 2 3 , row2 column 1 , 2 , column 2 0 , end matrix
20. [ 1 2 5 3 1 0 1 2 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 2 , column 3 5 , row2 column 1 , 3 , column 2 1 , column 3 0 , row3 column 1 , 1 , column 2 2 , column 3 1 , end matrix
21. [ 1 4 0 2 3 5 0 1 0 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 4 , column 3 0 , row2 column 1 , 2 , column 2 3 , column 3 5 , row3 column 1 , 0 , column 2 1 , column 3 0 , end matrix
22. [ − 2 4 1 3 0 − 1 1 2 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , negative 2 , column 2 4 , column 3 1 , row2 column 1 , 3 , column 2 0 , column 3 negative 1 , row3 column 1 , 1 , column 2 2 , column 3 1 , end matrix
23. [ 2 3 0 1 2 5 7 0 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 2 , column 2 3 , column 3 0 , row2 column 1 , 1 , column 2 2 , column 3 5 , row3 column 1 , 7 , column 2 0 , column 3 1 , end matrix

Graphing Calculator Evaluate the determinant of each 3 × 3 3 times 3  matrix.

24. [ 1 0 0 0 1 0 0 0 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 0 , column 3 0 , row2 column 1 , 0 , column 2 1 , column 3 0 , row3 column 1 , 0 , column 2 0 , column 3 1 , end matrix
25. [ 0 − 2 − 3 1 2 4 − 2 0 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 0 , column 2 negative 2 , column 3 negative 3 , row2 column 1 , 1 , column 2 2 , column 3 4 , row3 column 1 , negative 2 , column 2 0 , column 3 1 , end matrix
26. [ 12.2 13.3 9 1 − 4 − 17 21.4 − 15 0 ] . matrix with 3 rows and 3 columns , row1 column 1 , 12.2 , column 2 13.3 , column 3 9 , row2 column 1 , 1 , column 2 negative 4 , column 3 negative 17 , row3 column 1 , 21.4 , column 2 negative 15 , column 3 0 , end matrix

27. Use the map to determine the approximate area of the Bermuda Triangle.

    

Determine whether each matrix has an inverse. If an inverse matrix exists, find it.

28. [ 2 − 1 1 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 1 , row2 column 1 , 1 , column 2 0 , end matrix
29. [ 2 3 1 1 ] . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 3 , row2 column 1 , 1 , column 2 1 , end matrix
30. [ 2 3 2 4 ] . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 3 , row2 column 1 , 2 , column 2 4 , end matrix
31. [ 1 3 2 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 3 , row2 column 1 , 2 , column 2 0 , end matrix
32. [ 6 − 8 − 3 4 ] . matrix with 2 rows and 2 columns , row1 column 1 , 6 , column 2 negative 8 , row2 column 1 , negative 3 , column 2 4 , end matrix
33. [ 4 8 − 3 − 2 ] . matrix with 2 rows and 2 columns , row1 column 1 , 4 , column 2 8 , row2 column 1 , negative 3 , column 2 negative 2 , end matrix
34. [ − 1.5 3 2.5 − 0.5 ] . matrix with 2 rows and 2 columns , row1 column 1 , negative 1.5 , column 2 3 , row2 column 1 , 2.5 , column 2 negative 0.5 , end matrix
35. [ 1 − 2 3 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 negative 2 , row2 column 1 , 3 , column 2 0 , end matrix
36. Error Analysis A student wrote [ 1 1 2 1 3 1 4 ] . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 1 half , row2 column 1 , 1 third , column 2 1 fourth , end matrix  as the inverse of [ 1 2 3 4 ] , . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 2 , row2 column 1 , 3 , column 2 4 , end matrix . comma  What mistake did the student make? Explain your reasoning.

See Problem 5.

37. Use the coding matrix in Problem 5 to encode the phone number (555) 358-0001.

Table of Contents