Problem 13.13
Find, if possible, the inverse of each of the matrices(i)A=[
1 − 13
221
04 − 5]
(ii)B=[
2 − 11
203
1 − 10](i) First check the determinant ofA:A=∣
∣
∣
∣
∣1 − 13
221
04 − 5∣
∣
∣
∣
∣= 1∣
∣
∣
∣21
4 − 5∣
∣
∣
∣−(−^1 )∣
∣
∣
∣21
0 − 5∣
∣
∣
∣+^3∣
∣
∣
∣22
04∣
∣
∣
∣=− 10 − 4 + 2 (− 5 )− 0 × 1 + 3 ( 2 × 4 − 0 × 2 )
= 0
SoAissingular– its determinant is zero, and so its inverse does not exist.(ii) Hoping for better luck withBwe have
|B|=∣
∣
∣
∣
∣2 − 11
203
1 − 10∣
∣
∣
∣
∣= 2 ( 3 )+(− 3 )+(− 2 )
= 1
SotheinverseofBexists and we are not wasting our time findingAdjB.AdjB=
∣
∣
∣
∣03
− 10∣
∣
∣
∣ −∣
∣
∣
∣23
10∣
∣
∣
∣∣
∣
∣
∣20
1 − 1∣
∣
∣
∣−∣
∣
∣
∣− 11
− 10∣
∣
∣
∣∣
∣
∣
∣21
10∣
∣
∣
∣ −∣
∣
∣
∣2 − 1
1 − 1∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣− 11
03∣
∣
∣
∣ −∣
∣
∣
∣21
23∣
∣
∣
∣∣
∣
∣
∣2 − 1
20∣
∣
∣
∣
T=[ 33 − 2
− 1 − 11
− 3 − 4 − 2]T=[ 3 − 1 − 3
3 − 1 − 4
− 212]As an exercise you might like to check thatBAdjB=|B|Iat this stage.
So we now haveB−^1 =AdjB
|B|=AdjB=[ 3 − 1 − 3
3 − 1 − 4
− 212]