Proposition

Suppose the coefficient matrix $\Mtrx{A}$ of the system of $n$ linear equations in $n$ variables

\[ \begin{array}{rcccrcr} \colorbox{lightgreen}{$a_{11}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{1n}$} {\color{red} x_n} & = & c_1 \\ \vdots\ \ \ & & & & \vdots\ \ \ & & \vdots\ \ \\ \colorbox{lightgreen}{$a_{n1}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{nn}$} {\color{red} x_n} & = & c_n \end{array} \]

is invertible. Then this system has the unique solution

\[ {\color{red}\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}}\ =\ A^{-1} \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} \]

Proof

We know that the given system of linear equations is equivalent to the matrix equation.

\[\Mtrx{A}{\color{red} \Mtrx{X}} = \Mtrx{C}\]

Now we can use properties of matrix multiplication to find that this matrix equation is equalent to

\[ \begin{array}{rcl} \Mtrx{A}{\color{red} \Mtrx{X}} & = & \Mtrx{C} \\ \Mtrx{A}^{-1}(\Mtrx{A}{\color{red} \Mtrx{X}}) & = & \Mtrx{A}^{-1}\Mtrx{C} \\ (\Mtrx{A}^{-1}\Mtrx{A}){\color{red} \Mtrx{X}} & = & \Mtrx{A}^{-1}\Mtrx{C} \\ \IdMtrx{n}{\color{red} \Mtrx{X}} & = & \Mtrx{A}^{-1}\Mtrx{C} \\ {\color{red} \Mtrx{X}} & = & \Mtrx{A}^{-1}\Mtrx{C} \end{array} \]

The last identity means that the given system of linear equations has a unique solution, and this solution is given by the matrix product on the right as claimed.