I've been struggling for hours trying to solve this: For what value of k does the following system of linear equations have infinitely many solutions?
$$x+y+kz=3$$
$$x+ky+z=-7$$
$$kx+y+z=4$$
[Math] For what value of k does the following system of linear equations have infinitely many solutions
linear algebra
Best Answer
The system of equations can be written as a matrix:$$\left[ \begin{array}{ccc|c} 1&1&k&3\\ 1&k&1&-7\\k&1&1&4 \end{array} \right] $$ Find the determinant of the coefficient matrix and we get $-k^3+3k-2=0$
In order for the solution to be infinitely many solutions, the determinant must be 0. If det(A) = 0, that means at least one of the rows is a linear combination of the other rows.
So $-k^3+3k-2=0,k=1$