The sum of three numbers is 6. When second number is subtracted
from thrice the sum of first and third number, we get number 10. Four
times the third number is subtracted from five times the sum of first
and second number, the result is 3. Using above information, find
these three numbers by matrix method.
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Let the numbers be x,y and z
According to question, x+y+z=6 ...... (i)
3(x+z)−y=10
or 3x−y+3z=10 ........ (ii)
Also, 5(x+y)−4z=3
or 5x+5y−4z=3 ........ (iii)
Thus,
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
1
3
5
1
−1
5
1
3
−4
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
6
10
3
⎦
⎥
⎥
⎤
Applying R
3
→R
3
−5R
1
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
1
3
0
1
−1
0
1
3
−9
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
6
10
−27
⎦
⎥
⎥
⎤
Thus, the equations reduce to
x+y+z=6 ..... (iv)
3x−y+3z=10 ..... (v)
−9z=−27⇒z=3 .... (vi)
Adding (iv) and (v), we get
4x+4z=16⇒x+z=4 ..... (vii)
Substitute value of z from (vi) in (vii),
x+3=4⇒x=1
Substitute the values of x and z in (iv), we get
1+y+3=6⇒y=2
Hence, the numbers are 1,2 and 3.
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