Exercise 1.1
1. Find the minors and cofactors of
all the elements of the following
determinants.
go
5 20
0
51
1 - 3 2
(ii) 4 -12
3 5 2
0
-1
Answers
(i)
∣
∣
∣
∣
∣
∣
∣
∣
1
0
0
0
1
0
0
0
1
∣
∣
∣
∣
∣
∣
∣
∣
(ii)
∣
∣
∣
∣
∣
∣
∣
∣
1
3
0
0
5
1
4
−2
2
∣
∣
∣
∣
∣
∣
∣
∣
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VIDEO EXPLANATION
ANSWER
(i) A=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
The elements of the minor of the matrix A will be given by
M
1,1
=
∣
∣
∣
∣
∣
∣
1
0
0
1
∣
∣
∣
∣
∣
∣
=1j
M
1,2
=
∣
∣
∣
∣
∣
∣
0
0
0
1
∣
∣
∣
∣
∣
∣
=0
M
1,3
=
∣
∣
∣
∣
∣
∣
0
0
1
0
∣
∣
∣
∣
∣
∣
=0
M
2,1
=
∣
∣
∣
∣
∣
∣
0
0
0
1
∣
∣
∣
∣
∣
∣
=0
M
2,2
=
∣
∣
∣
∣
∣
∣
1
0
0
1
∣
∣
∣
∣
∣
∣
=1
M
2,3
=
∣
∣
∣
∣
∣
∣
1
0
0
0
∣
∣
∣
∣
∣
∣
=0
M
3,1
=
∣
∣
∣
∣
∣
∣
0
1
0
0
∣
∣
∣
∣
∣
∣
=0
M
3,2
=
∣
∣
∣
∣
∣
∣
1
0
0
0
∣
∣
∣
∣
∣
∣
=0
M
3,3
=
∣
∣
∣
∣
∣
∣
1
0
0
1
∣
∣
∣
∣
∣
∣
=1
Hence, M=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
The cofactor matrix will be given by C
i,j
=(−1)
i+j
M
i,j
∴C=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
(ii) B=
⎣
⎢
⎢
⎡
1
3
0
0
5
1
4
−2
2
⎦
⎥
⎥
⎤
M
1,1
=
∣
∣
∣
∣
∣
∣
5
1
−2
2
∣
∣
∣
∣
∣
∣
=12
M
1,2
=
∣
∣
∣
∣
∣
∣
3
0
−2
2
∣
∣
∣
∣
∣
∣
=6
M
1,3
=
∣
∣
∣
∣
∣
∣
3
0
5
1
∣
∣
∣
∣
∣
∣
=3
M
2,1
=
∣
∣
∣
∣
∣
∣
0
1
4
2
∣
∣
∣
∣
∣
∣
=−4
M
2,2
=
∣
∣
∣
∣
∣
∣
1
0
4
2
∣
∣
∣
∣
∣
∣
=2
M
2,3
=
∣
∣
∣
∣
∣
∣
1
0
0
1
∣
∣
∣
∣
∣
∣
=1
M
3,1
=
∣
∣
∣
∣
∣
∣
0
5
4
−2
∣
∣
∣
∣
∣
∣
=−20
M
3,2
=
∣
∣
∣
∣
∣
∣
1
3
4
−2
∣
∣
∣
∣
∣
∣
=−14
M
3,3
=
∣
∣
∣
∣
∣
∣
1
3
0
5
∣
∣
∣
∣
∣
∣
=5
Hence, M=
⎣
⎢
⎢
⎡
12
−4
−20
6
2
−14
3
1
5
⎦
⎥
⎥
⎤
The cofactor matrix will be given by C
i,j
=(−1)
i+j
M
i,j
∴C=
⎣
⎢
⎢
⎡
12
4
−20
−6
2
14
3
−1
5
⎦
⎥
⎥
⎤
Explanation:
ANSWER
(i) Given equation is,
4x
2
=3x
Simplifying the equation,
⇒4x
2
−3x=0
⇒x(4x−3)=0
⇒x=0 or 4x−3=0
⇒x=0 or 4x=3
⇒x=0 or x=
4
3
∴ Value of x=0,
4
3
(ii) Given equation is,
2
(x
2
−5x)
=0
Simplifying the equation,
⇒x
2
−5x=0
⇒x(x−5)=0
⇒x=0 or x−5=0
⇒x=0 or x=5
∴ Value of x=0,5
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