Inverse of a matrix Find the inverse of the matrix A, by adjoint matrix: 1 3 1. 1 3 -3 -2 – 4 – 4 1
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Answers
Answer:
⎡
1
1
2
2
1
4
3
5
7
⎦
⎥
⎥
⎤
The co-factor matrix
C
11
=(−1)
1+1
M
11
=
∣
∣
∣
∣
∣
∣
1
4
5
7
∣
∣
∣
∣
∣
∣
=7−20=−13
C
12
=(−1)
1+2
M
12
=−
∣
∣
∣
∣
∣
∣
1
2
5
7
∣
∣
∣
∣
∣
∣
=−(7−10)=3
C
13
=(−1)
1+3
M
13
=
∣
∣
∣
∣
∣
∣
1
2
1
4
∣
∣
∣
∣
∣
∣
=4−2=2
C
21
=(−1)
2+1
M
21
=−
∣
∣
∣
∣
∣
∣
2
4
3
7
∣
∣
∣
∣
∣
∣
=−(14−12)=−2
C
22
=(−1)
2+2
M
22
=
∣
∣
∣
∣
∣
∣
1
2
3
7
∣
∣
∣
∣
∣
∣
=7−6=1
C
23
=(−1)
2+3
M
23
=−
∣
∣
∣
∣
∣
∣
1
2
2
4
∣
∣
∣
∣
∣
∣
=−(4−4)=0
C
31
=(−1)
3+1
M
31
=
∣
∣
∣
∣
∣
∣
2
1
3
5
∣
∣
∣
∣
∣
∣
=10−3=7
C
32
=(−1)
3+2
M
32
=−
∣
∣
∣
∣
∣
∣
1
1
3
5
∣
∣
∣
∣
∣
∣
=−(5−3)=−2
C
33
=(−1)
3+3
M
33
=
∣
∣
∣
∣
∣
∣
1
1
2
1
∣
∣
∣
∣
∣
∣
=1−2=−1
Since the transpose of the co-factor matrix of A is adjA
⇒ Adj(A)
=
∣
∣
∣
∣
∣
∣
∣
∣
−13
−2
7
3
1
−2
2
0
−1
∣
∣
∣
∣
∣
∣
∣
∣
T
=
∣
∣
∣
∣
∣
∣
∣
∣
−13
3
2
−2
1
0
7
−2
−1
∣
∣
∣
∣
∣
∣
∣
∣
Since detA from the above matrix A
=1(7−20)−2(7−10)+3(4−2)=−13+6+6=−1
=0
Hence inverse of A
−1
exists.
A
−1
=
∣A∣
adjA
=−
∣
∣
∣
∣
∣
∣
∣
∣
−13
3
2
−2
1
0
7
−2
−1
∣
∣
∣
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
∣
∣
∣
13
−3
−2
2
−1
0
−7
2
1
∣
∣
∣
∣
∣
∣
∣
∣
Answer:
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Step-by-step explanation:
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