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$$x_1 + x_2 + x_3 = a$$

$$2x_1 - x_2 + 3x_3 = b$$

$$4x_1 + x_2 + 9x_3 = c$$

Now, looking at the solution of this, the author does the following:

The matrix of the coefficients is invertible because $det(E) \ne 0$ (where E is the matrix). Therefore the linear system has only one solution.

Can someone elaborate of the above solution? I do not understand this deduction.

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    $\begingroup$ @ChristianF I think the OP is using != to mean $\neq$ This comes from the C programming language. $\endgroup$ – saulspatz May 11 '18 at 19:44
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See the invertible matrix theorem. http://mathworld.wolfram.com/InvertibleMatrixTheorem.html

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When the matrix is not singular, i.e. $\det(A)\neq 0$, the columns of $A$ are linearly independent and thus, since $A\vec x$ represents a combination of column vectors which are a basis and span $R^3$, we have that $$A\vec x=b$$

has always one and only one solution.

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  • $\begingroup$ What has this got to do with a question? $\endgroup$ – Aqua May 11 '18 at 19:47
  • $\begingroup$ @ChristianF osp...I read wrongly! $\endgroup$ – user May 11 '18 at 19:48
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Multiplying the first equation by $-2$ and adding to the second: $$-3x_2+x_3=b-2a$$ Multiplying the first by $-4$ and adding to the third we get $$-3x_2+5x_3=c-4a$$ From this equations we get $$4x_3=-2a-b+c$$

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    $\begingroup$ What has this got to do with a question? $\endgroup$ – Aqua May 11 '18 at 19:47
  • $\begingroup$ This answers the question which is given above! $\endgroup$ – Dr. Sonnhard Graubner May 11 '18 at 19:49
  • $\begingroup$ Can you write down this question, please! $\endgroup$ – Aqua May 11 '18 at 19:49
  • $\begingroup$ It proves that x_3 of the solution is unique, and then x_2 and x_1 will be unique as well. $\endgroup$ – primes.against.humanity May 11 '18 at 19:53
  • $\begingroup$ Yes and the system has only one solution! $\endgroup$ – Dr. Sonnhard Graubner May 11 '18 at 19:54
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If you write the system as a matrix equation

\begin{align*} \left( \begin{matrix} 1 &1 &1\\ 2 &-1 &3\\ 4 &1 &9 \end{matrix} \right) \left( \begin{matrix} x_1\\ x_2\\ x_3 \end{matrix}\right) = \left( \begin{matrix} a\\ b\\ c \end{matrix}\right), \end{align*}

then the determinant of your coefficient matrix is non-zero, so there is a unique solution, which is the right hand side of your equation times the inverse of the coefficient matrix:.

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Theorem: If $A$ is an invertible $n\times n$ matrix, then $\forall b\in\Bbb R^n$, the system $Ax=b$ has a unique solution.

You must know this theorem by heart!

Proof: Indeed, if $x_0$ is a solution, then $$Ax_0 = b\Rightarrow A^{-1}(Ax_0) = A^{-1}b \Rightarrow (A^{-1}A)x_0 = Ix_0 = x_0 =A^{-1}b$$ so this proves that the only possible solution is $A^{-1}b$.

Now, $A^{-1}b$ is really a solution since $A(A^{-1}b) = (AA^{-1})b = Ib = b$.

Hence $Ax=b$ has one and only one solution if $A$ is invertible.$\;\;\blacksquare$

Remark: Of course, this is only a possible proof, and other answers above are also good.

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