122 CHAPTER 6. DETERMINANTS
9. Find the determinant using row operations.
det
1 4 1 23 2 −2 3−1 0 3 32 1 2 −2
10. Verify an example of each property of determinants found in Theorems 6.1.23 -
6.1.25 for 2×2 matrices.
11. An operation is done to get from the first matrix to the second. Identify what wasdone and tell how it will affect the value of the determinant.(
a bc d
),
(a cb d
)
12. An operation is done to get from the first matrix to the second. Identify what wasdone and tell how it will affect the value of the determinant.(
a bc d
),
(c da b
)
13. An operation is done to get from the first matrix to the second. Identify what wasdone and tell how it will affect the value of the determinant.(
a bc d
),
(a b
a+ c b+d
)
14. An operation is done to get from the first matrix to the second. Identify what wasdone and tell how it will affect the value of the determinant.(
a bc d
),
(a b2c 2d
)
15. An operation is done to get from the first matrix to the second. Identify what wasdone and tell how it will affect the value of the determinant.(
a bc d
),
(b ad c
)
16. 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.
17. Show det(aA) = an det(A) where here A is an n×n matrix and a is a scalar.
18. Illustrate with an example of 2×2 matrices that the determinant of a product equalsthe product of the determinants.