Find the value(s) of k for which the pair of linear equations kx + y = k^{2} and x + ky = 1 have infinitely many solutions.

### Answer

For pair of equations kx + 1y = k^{2} and 1x + ky = 1

We have:

a_{1}/a_{2} = k/1

b_{1}/b_{2} = 1/k

c_{1}/c_{2} = k^{2}/1

For infinitely many solutions,

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

k/1 = 1/k

k^{2} = 1

k = 1, -1 ... (i)

1/k = k^{2}/1

k^{3} = 1

k = 1 ... (ii)

From (i) and (ii),

k = 1