# Thread: Properties of determinants

1. ## Properties of determinants

Determine all values of k for which the given system has an infinite number of solutions:
1x1 + 2x2 + x3 = kx1
2x1 + 1x2 + 1x3 = kx2
1x1 +1 x2 + 2x3 = kx3

I know it is probably as simple as finding the determinant and then applying the properties of the determinant, but I am thrown off by the fact that k is in the solutions column. Please help! Thanks in advance.

2. All of the numbers following the x's are supposed to be subscripts, but it didn't transfer properly into the post. Sorry bout that . . .

3. Originally Posted by dan213
All of the numbers following the x's are supposed to be subscripts, but it didn't transfer properly into the post. Sorry bout that . . .
$\displaystyle \begin{array}{rrrrrrr}x_1&+&2x_2&+&x_3&=&kx_1\\2x_ 1&+&x_2&+&x_3&=&kx_2\\x_1&+&x_2&+&2x_3&=&kx_3\end{ array}\Longleftrightarrow$ $\displaystyle \begin{array}{rrrrrrr}(1-k)x_1&+&2x_2&+&x_3&=&0\\2x_1&+&(1-k)x_2&+&x_3&=&0\\x_1&+&x_2&+&(2-k)x_3&=&0\end{array}$

So we have a square linear homogeneous system, and it has a unique solution iff the coefficients'

matrix's determinant doesn't equal zero, so when this determinant is zero there's more than one

unique solution, and if the definition field is infinite then there are infinite solutions.

BTW, the determinant above is $\displaystyle -(k^3-4k^2-k+4)$ , and one of the values for which it equals zero is $\displaystyle k=-1$ ...

Tonio

4. Thanks Tonio, excellent answer.