1016-331 RIT, 20091 1
Linear Algebra I 1016-331
Quiz 6 Solution
1. Find the adjoint and the inverse of the matrix
A=
a0 0
1a0
0 1 a
.
We compute:
AT=
det ·a0
1a¸−det ·1 0
0a¸det ·1a
0 1 ¸
−det ·0 0
1a¸det ·a0
0a¸−det ·a0
0 1 ¸
det ·0 0
a0¸−det ·a0
1 0 ¸det ·a0
1a¸
T
=
a20 0
−a a20
1−a a2
.
Check:
AAT=
a0 0
1a0
0 1 a
a20 0
−a a20
1−a a2
=
a30 0
0a30
0 0 a3
.
So the inverse is (for a6= 0):
A−1=
1/a 0 0
−1/a21/a 0
1/a3−1/a21/a
.
2. Find the solution of the linear system
x+ 3y−z= 4
x−y+ 2z= 5
2x+ 2y−z= 3.
We compute (Cramer’s rule):
x=
det
4 3 −1
5−1 2
3 2 −1
det
1 3 −1
1−1 2
2 2 −1
=8
8=1, y =
det
1 4 −1
1 5 2
2 3 −1
det
1 3 −1
1−1 2
2 2 −1
=16
8=2,
z=
det
1 3 4
1−1 5
2 2 3
det
1 3 −1
1−1 2
2 2 −1
=24
8=3.