Question

Solve the system of equations by using the inverse of the coefficient matrix of the equivalent matrix equation.

Answer 1

The solution for the given system of equations x + 8y = -37, 4x + 8y = -52 is $\left[\begin{array}{ccc}&5&\\\\&4&\end{array}\right]$

Given,

System of equations as,

x + 8y = -37

4x + 8y = -52

We have to solve this by using the inverse of coefficient matrix of the equivalent matrix equation.

That is,

$A=\left[\begin{array}{ccc}a&&b\\\\c&&d\end{array}\right]$

$A^{-1} =\frac{1}{ad -bc} \left[\begin{array}{ccc}d&&-b\\\\-c&&a\end{array}\right]$

Now we can solve the equations.

Here we have,

x + 8y = -37

4x + 8y = -52

Now in matrix form,

$\left[\begin{array}{ccc}1&&8\\\\4&&8\end{array}\right]$  $\left[\begin{array}{ccc}&x&\\\\&y&\end{array}\right]$ $=\left[\begin{array}{ccc}&-37&\\\\&-52&\end{array}\right]$

      A               X                  B

We know that,

$A^{-1} =\frac{1}{ad -bc} \left[\begin{array}{ccc}d&&-b\\\\-c&&a\end{array}\right]$

Therefore,

$A^{-1} = \frac{1}{(1X8)-(4X8)} \left[\begin{array}{ccc}8&&-8\\\\-4&&1\end{array}\right]$

       $=\frac{1}{8-32} \left[\begin{array}{ccc}8&&-8\\\\-4&&1\end{array}\right]$

       $=\frac{1}{-24} \left[\begin{array}{ccc}8&&-8\\\\-4&&1\end{array}\right]$

       $=\left[\begin{array}{ccc}\frac{-8}{24} &&\frac{8}{24} \\\\\frac{4}{24} &&\frac{-1}{24} \end{array}\right]$

Then,

$\left[\begin{array}{ccc}&x&\\\\&y&\end{array}\right] =$$=\left[\begin{array}{ccc}\frac{-8}{24} &&\frac{8}{24} \\\\\frac{4}{24} &&\frac{-1}{24} \end{array}\right]$  $\left[\begin{array}{ccc}&\frac{37}{52} &\\\\\end{array}\right]$

                $=\frac{1}{24} \left[\begin{array}{ccc}-8&&8\\\\4&&-1\end{array}\right]$$\left[\begin{array}{ccc}&\frac{37}{52} &\\\\\end{array}\right]$

                $=\frac{1}{24} \left[\begin{array}{ccc}(-8X37)+(8X52)\\\\(4X37)+(-1X52)\end{array}\right]$

               $=\frac{1}{24} \left[\begin{array}{ccc}-296+416\\\\148-52\end{array}\right]$

               $=\frac{1}{24} \left[\begin{array}{ccc}120\\\\96\end{array}\right]$

               $=\left[\begin{array}{ccc}\frac{120}{24} \\\\\frac{96}{24} \end{array}\right]$

              $=\left[\begin{array}{ccc}5\\\\4\end{array}\right]$

That is $\left[\begin{array}{ccc}x\\\\y\end{array}\right] =\left[\begin{array}{ccc}5\\\\4\end{array}\right]$

Learn more about matrix equations here: brainly.com/question/27799804

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The question is incomplete. Completed question is given below:

Solve The System Of Equations By Using The Inverse Of The Coefficient Matrix Of The Equivalent Matrix Equation.

x + 8y = -37

4x + 8y = -52