Page 187 - 35Linear Algebra

8.4 Properties of the Determinant 187

Example 103 Let’s compute the determinant of

 
1 2 3
M =  4 5 6 
7 8 9

using expansion by minors:

5 6 4 6 4 5
det M = 1 det − 2 det + 3 det
8 9 7 9 7 8
= 1(5 · 9 − 8 · 6) − 2(4 · 9 − 7 · 6) + 3(4 · 8 − 7 · 5)

= 0

1
Here, M −1 does not exist because det M = 0.

Example 104 Sometimes the entries of a matrix allow us to simplify the calculation
 
1 2 3
of the determinant. Take N =  4 0 0 . Notice that the second row has many

7 8 9
zeros; then we can switch the first and second rows of N before expanding in minors
to get:

   
1 2 3 4 0 0
det  4 0 0  = − det  1 2 3 
7 8 9 7 8 9

2 3
= −4 det
8 9
= 24

Example

Since we know how the determinant of a matrix changes when you perform
row operations, it is often very beneficial to perform row operations before
computing the determinant by brute force.

1
A fun exercise is to compute the determinant of a 4 × 4 matrix filled in order, from
left to right, with the numbers 1, 2, 3, . . . , 16. What do you observe? Try the same for a
5 × 5 matrix with 1, 2, 3, . . . , 25. Is there a pattern? Can you explain it?

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