where
=∣
∣
∣
∣
∣a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣
∣
∣
∣
∣Again, note the pattern– denominators are always the same and equal to the determinant
of the coefficients,. The numerator ofxi is obtained by replacing the column ofxi
coefficients inby the column of right-hand sides.
Problem 13.11
Solve the system of linear equations3 xY 2 y−z= 0
2 x−yYz= 1
x−yY 2 z=− 1Using Cramer’s rule we have
x=∣
∣
∣
∣
∣02 − 1
1 − 11
− 1 − 12∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣32 − 1
2 − 11
1 − 12∣ ∣ ∣ ∣ ∣ =0∣
∣
∣
∣− 11
− 12∣
∣
∣
∣−^2∣
∣
∣
∣11
− 12∣
∣
∣
∣−^1∣
∣
∣
∣1 − 1
− 1 − 1∣
∣
∣
∣3∣
∣
∣
∣− 11
− 12∣
∣
∣
∣−^2∣
∣
∣
∣21
12∣
∣
∣
∣−^1∣
∣
∣
∣2 − 1
1 − 1∣
∣
∣
∣=− 2 ( 2 + 1 )−(− 1 − 1 )
3 (− 2 + 1 )− 2 ( 4 − 1 )−(− 2 + 1 )=− 4
− 8=1
2y=∣
∣
∣
∣
∣30 − 1
211
1 − 12∣
∣
∣
∣
∣− 8=3 ( 2 + 1 )− 1 (− 2 − 1 )
− 8=9 + 3
− 8=−3
2z=∣
∣
∣
∣
∣320
2 − 11
1 − 1 − 1∣
∣
∣
∣
∣
− 8