311.2.2.6 use inverse matrix to solve system of equations

[karush]

$\tiny{311.2.2.6}$
Use the inverse to solve the system
$\begin{array}{rrrrr}
7x_1&+3x_2&=-9\\
-2x_1&+x_2&=10
\end{array}$

Spoiler: solution

$A^{-1}b=x$
$\begin{bmatrix}
\frac{1}{13}&-\frac{3}{13} \\ \\ \frac{2}{13}& \frac{7}{13} \end{bmatrix}
\begin{bmatrix}
-9\\ \\ 10
\end{bmatrix}
=
\begin{bmatrix}
-3\\ \\ 4
\end{bmatrix}$

the thing I could not get here without a calculator is $A^{-1}$

 

[topsquark]

Given \(\displaystyle A = \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )\)

When all else fails:
\(\displaystyle A^{-1} = \dfrac{1}{ |A| } \left ( \begin{matrix} d & -b \\ -c & a \end{matrix} \right )\)

I have this one memorized.

-Dan

 

[karush]

ok i think i had a and switched