I've got to find the y value using Cramer's rule from the following set of equations:
$$ 4x – 2y – 3z = 5 $$
$$ -2x – 4y + z = 21 $$
$$ 8x – y – 2z = 7 $$
I'm stuck on the part that requires finding the determinant of the top matrix (which should be 370, but I get different answers each time is reduce a different number to 0).
My working out is as follows:
I can't figure out why the determinant changes, but there must be some relation since they're multiples of 370?
y is -5 when the determinant is 370, but I can't get a determinant of 370.
Best Answer
Adding a scalar multiple of one row (or column) to another row (or column) does not change the determinant so in your third attempt, you can indeed add twice the second row to the first row and you obtain the correct value of $370$.
However, in your first and second attempt, you change the second row not only by adding another row (the first), but also the second row itself. This changes the determinant: multiplying a row (or column) with a factor $k$, results in the determinant being multiplied by that same factor $k$ as well, hence the $740$ (2 times $R_2$ ) and $1110$ (3 times $R_2$).
As a side note, I would focus on the last column: $$\begin{vmatrix} 4 & 5 & \color{red}{-3} \\ -2 & 21 & \color{blue}{1} \\ 8 & 7 & \color{red}{-2} \end{vmatrix} \underset{R_3 \to R_1+2R_2}{\overset{R_1 \to R_1+3R_2}{\longrightarrow}} \begin{vmatrix} -2 & 68 & \color{red}{0} \\ -2 & 21 & \color{blue}{1} \\ 4 & 49 & \color{red}{0} \end{vmatrix}= - \left( -2 \cdot 49 - 4 \cdot 68 \right) = 370 $$