GATE Linear Algebra

Question

For what values of $\lambda$ the system of equations will have $2$ linear independent solutions -

$x + y + z = 0$

$(\lambda + 1) y + (\lambda + 1) z = 0$

($\lambda^{2}- 1) z = 0$

 

Now the problem i'm facing is if there is $2$ Linear independent solutions means there will be $2$ non - zero rows in row reduced echelon form so answer will be $\lambda = -1$, $1$ but actual answer given is $\lambda = -1$ and explanation was given that rank of matrix will be $1$.

Can anyone please explain in simple why the rank of matrix should be $1$ if we need $2$ Linear Independent solution.

Thankyou.

Comments

The number of Linearly Independent Solution of  'M' homogenous linear equations in 'N' variables
is N-R (R is rank)
N-R=2
3-R=2
R=1

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You got confused between independent solution and linearly independent vectors/columns.

Number of linearly independent vectors/columns = non-zero rows =rank

Number of independent solution = n-rank

Answer 1

no. of Linearly Independent solutions=n-r

where n is no. of variables and r is rank of matrix

so 2=3-r

r=1

now if you put value of lambda =-1 in matrix You will get rank of matrix as 1

thus lambda =-1 is the solution