i have the resolution of this exercise by the Gauss-jordan's method, but i want to know how to solve it by determinants.
The exercise asks us to discuss and solve the following system: {λx + y − z = −1 {λx + λz = λ {x + y − λz = 0
I thought about using the Gauss-JOrdan's theorem as I said, by which rank(A)=rank(A') for the system to be compatible. But when I tried to calculate the rank of A it results in zero and I couldn't find any smaller non-nule determinant of A.
Is that even the way to solve it? Finding smaller non-nule determinants?
i already solved it, it was easier than i thought, the only thing you have to do is the determinant of the non ampliated matrix, which means the matrix formed by all lines except the last column. It results in zero, as i already knew, but now the seocond step is to calculate the determinant of the same matrix, but substituting whichever column by the one you didn't use before, then you equal it to zero, and you have obtained the values of lambda for wich the ampliated and non ampliated matrix share the same rank, and by Rouche-Frobenius's theorem you know in that case they're compatible.