GATE Mechanical Engineering (ME) 2019 Shift 1 Solved Paper

Question : 53 of 65

Marks: +1, -0

The set of equations

x+y+z=1

a x-a y+3 z=5

5 x-3 y+a x=6

has infinite solutions, if

{a}=

Solution:

The coefficient matrix is given by

[\table 1 , 1 , 1 ; a , -a , 3 ; 5 , -3 , a]

And the augmented matrix is given by

(\table 1 , 1 , 1 , 1 ; a , -a , 3 , 5 ; 5 , -3 , a,6)

For infinite solution, Rank(coefficient matrix) = Rank(augmented matrix)

<

Order of matrix(3)

(\table 1 , 1 , 1 , 1 ; a , -a , 3 , 5 ; 5 , -3 , a , 6)

R_{2} → R_{2}-a R_{1}, R_{3} → R_{3}-5 R_{1}

(\table 1 , 1 , 1 , 1 ; 0 , -2 a , 3-a , 5-a ; 0 , -8 , a-5 , 1)

R_{3} → a R_{3}-4 R_{2}

(\table 1 , 1 , 1 , 1 ; 0 , -2 a , 3-a , 5-a ; 0 , 0 , a^{2}-a-12 , 5 a-20)

For Rank(coefficient matrix) = Rank(augmented matrix

=2

a^{2}-a-12=0 \;\text " or "\; a=-3,4

And

5 a-20=0

a=4

So,

a=4

is the correct answer

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