|
- Find the determinants of these matrices. Show your work.
| a. |
 |
| b. |
 |
| c. |
 |
| d. |
 |
| e. |
 |
| f. |
 |
| g. |
 |
| h. |
 |
| i. |
 |
- Use Cramer's rule to solve these systems of equations:
-
| 2x1 |
+ |
3x2 |
- |
5x3 |
|
= |
|
-11 |
| -4x1 |
- |
x2 |
+ |
3x3 |
|
= |
|
3 |
| 3x1 |
- |
2x2 |
+ |
x3 |
|
= |
|
7 |
-
| x1 |
- |
5x2 |
+ |
7x3 |
|
= |
|
-10 |
| |
|
9x2 |
+ |
2x3 |
|
= |
|
7 |
| x1 |
+ |
3x2 |
- |
x3 |
|
= |
|
6 |
- Find the inverse of matrix A using cofactors and determinants. Verify that you found the inverse by checking that I is the product matrix of AA-1 or A-1A. Remember, since A is square, you do not have to
check both because if AA-1 or A-1A is the identity matrix, then so is the other.
- Prove whether the following statements are true or false for 2 by 2
matrices. Remember that a counterexample establishes that a statement is
false.
-
-
-
-
Remark 16 In general, you may NOT assume that a statement is true for all matrices because it is true for 2 by 2 matrices, but for the examples in this question, those that are true for 2 by 2 matrices are true for all matrices
if the dimensions allow the operations to be performed.
- Show that the determinant of an upper triangular matrix is the
product of the diagonal entries.
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