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First of - sorry for my English.

My math teacher gave me three tasks to complete, but I can't complete the second one.

Here is the system: $$\begin{cases} x+y-z=3 \\ x-3y+2z=1 \\ 7x-4y+z=7\end{cases}$$

We have to complete them using Cramer's rule. Determinant result is 43 for me, but today when I asked teacher she said there should be 1. I can't figure it out. How it can be 1? It would be great if someone could explain how to get to 1 step by step.

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  • $\begingroup$ Don't use images unless they can't be explained otherwise. And matrices definitely can be rendered here. Besides, we can't know where you made a mistake. $\endgroup$ – vonbrand Oct 1 '15 at 14:31
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The coefficient matrix for the system $$\begin{cases} x+y-z=3 \\ x-3y+2z=1 \\ 7x-4y+z=7\end{cases}$$ is $$\pmatrix{1 & 1 & -1 \\ 1 & -3 & 2 \\ 7 & -4 & 1}$$

Let's find the determinant:

$$\begin{align}\left|\begin{array}{ccc} 1 & 1 & -1 \\ 1 & -3 & 2 \\ 7 & -4 & 1\end{array}\right| &= \left|\begin{array}{ccc} 1 & 1 & -1 \\ 0 & -4 & 3 \\ 0 & -11 & 8\end{array}\right| \\ &= \frac 13\left|\begin{array}{ccc} 1 & 1 & -1 \\ 0 & -12 & 9 \\ 0 & -11 & 8\end{array}\right| \\ &= \frac 13\left|\begin{array}{ccc} 1 & 1 & -1 \\ 0 & -1 & 1 \\ 0 & -11 & 8\end{array}\right| \\ &= -\frac 13\left|\begin{array}{ccc} 1 & 1 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & -3\end{array}\right| \\ &= -\frac 13(1)(1)(-3) \\ &= 1\end{align}$$


EDIT: Here's the cofactor expansion method of finding the determinant. I'll expand across the top row:

$$\begin{align}\left|\begin{array}{ccc} 1 & 1 & -1 \\ 1 & -3 & 2 \\ 7 & -4 & 1\end{array}\right| &= (1)\left|\begin{array}{cc} -3 & 2 \\ -4 & 1\end{array}\right|-(1)\left|\begin{array}{cc} 1 & 2 \\ 7 & 1\end{array}\right|+(-1)\left|\begin{array}{cc} 1 & -3 \\ 7 & -4\end{array}\right| \\ &= (1)[-3\cdot 1 - 2\cdot(-4)] -(1)[1\cdot 1 - 2\cdot 7]+(-1)[1\cdot(-4)-(-3\cdot 7)] \\ &= 5+13-17 \\ &=1\end{align}$$

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  • $\begingroup$ We would use: = 1*(-3)*1+1*2*7+1*(-4)*(-1)... to get the determinant. So I'm even more lost after what you showed.. $\endgroup$ – mypoint Oct 1 '15 at 15:24
  • $\begingroup$ Ah. You haven't learned the relationship between the elementary row operations and the determinant are yet. OK. I'll edit in the cofactor expansion. Give me a minute. $\endgroup$ – user137731 Oct 1 '15 at 15:26
  • $\begingroup$ Yeah we have not :( Thanks a lot! :) $\endgroup$ – mypoint Oct 1 '15 at 15:33
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    $\begingroup$ Or here's Sarrus's rule. $\endgroup$ – user137731 Oct 1 '15 at 15:37
  • $\begingroup$ Huge thanks! :) Now it makes sense!!! $\endgroup$ – mypoint Oct 1 '15 at 15:40

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