Page 90 - A Course in Linear Algebra with Applications

74 Chapter Three: Determinants

1 1 1 1 1 1 1 1
0 1 2 3 0 1 2 3
D =
- 2 - 2 3 3 0 0 5 5
1 - 2 -2 - 3 0 - 3 -3 - 4
Next apply successively i? 4 + 3i? 2 and l/5i?3 to get

1 1 1 1
0 1 2 2
D = - 5
0 0 5 0 0 1 1
0 0 3 0 0 3 5

Finally, use of R4 — 3i?3 yields
1 1 1
n
D = -5 0 1 2 = -10.
0 0 1
0 0 0

Example 3.2.2
Use row operations to show that the following determinant is
identically equal to zero.

a + 2 b + 2 c + 2
x + 1 y + 1 z + 1
2x — a 2y — b 2z — c

Apply row operations R% + Ri and 2R2. The resulting
determinant is zero since rows 2 and 3 are identical.
Example 3.2.3
Prove that the value of the n x n determinant

2 1 0 • • 0 0 0
1 2 1 • • 0 0 0

0 0 0 • • 1 2 1
0 0 0 • • 0 1 2