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GATE Mechanical Engineering (ME) 2019 Shift 1 Solved Paper

Question : 53 of 65

Marks: +1, -0

x+y+z=1

a x - a y + 3 z = 5

5 x - 3 y + a x = 6

a =

Solution:

The coefficient matrix is given by

\begin{bmatrix} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{bmatrix}

And the augmented matrix is given by

\begin{pmatrix} 1 & 1 & 1 & 1 \\ a & -a & 3 & 5 \\ 5 & -3 & a & 6 \end{pmatrix}

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

<

Order of matrix(3)

\begin{pmatrix} 1 & 1 & 1 & 1 \\ a & -a & 3 & 5 \\ 5 & -3 & a & 6 \end{pmatrix}

R_2 \rightarrow R_2 - a R_1, R_3 \rightarrow R_3 - 5 R_1

\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & -2a & 3-a & 5-a \\ 0 & -8 & a-5 & 1 \end{pmatrix}

R_3 \rightarrow a R_3 - 4 R_2

\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & -2a & 3-a & 5-a \\ 0 & 0 & a^2 - a - 12 & 5a - 20 \end{pmatrix}

For Rank(coefficient matrix) = Rank(augmented matrix

=2

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

And

5a - 20 = 0

a = 4

So,

a = 4

is the correct answer

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