# Find the inverse of the following matrices: A=begin{bmatrix}2 & 3 -1 & -4 end{bmatrix} , B=begin{bmatrix}1 & -4&8 -3 & 5&76&2&-2 end{bmatrix}

Find the inverse of the following matrices: $A=\left[\begin{array}{cc}2& 3\\ -1& -4\end{array}\right],B=\left[\begin{array}{ccc}1& -4& 8\\ -3& 5& 7\\ 6& 2& -2\end{array}\right]$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

step 1
Ans: We have to find the inverse of the matrices:
(1) $A=\left[\begin{array}{cc}2& 3\\ -1& -4\end{array}\right]$
${A}^{-1}=\frac{adj\left(A\right)}{|A|}$
Step 2
$|A|=\left[\begin{array}{cc}2& 3\\ -1& -4\end{array}\right]=-8+3=-5$
$adj\left(A\right)=\left[\begin{array}{cc}-4& -3\\ 1& 2\end{array}\right]$
$\therefore {A}^{-1}=\frac{1}{5}\left[\begin{array}{cc}-4& -3\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}\frac{4}{5}& \frac{3}{5}\\ -\frac{1}{5}& -\frac{2}{5}\end{array}\right]$
(2)$B=\left[\begin{array}{ccc}1& -4& 8\\ -3& 5& 7\\ 6& 2& -2\end{array}\right]$
Step 3
${B}^{-1}=\frac{adj\left(B\right)}{|B|}$
So,$|B|=\left[\begin{array}{ccc}1& -4& 8\\ -3& 5& 7\\ 6& 2& -2\end{array}\right]=1\left(-10-14\right)+4\left(6-42\right)+8\left(-6-30\right)$
$=-24+4\left(-36\right)+8\left(-36\right)$
$=-456$
$adj\left(B\right)=\left[\begin{array}{ccc}-24& 8& -68\\ 36& -50& -31\\ -36& -26& -7\end{array}\right]$
$⇒{B}^{-1}=\frac{adj\left(B\right)}{|B|}=\frac{-1}{456}\left[\begin{array}{ccc}-24& 8& -68\\ 36& -50& -31\\ -36& -26& -7\end{array}\right]$
$=\left[\begin{array}{ccc}\frac{1}{19}& -\frac{1}{57}& \frac{17}{114}\\ -\frac{3}{38}& \frac{25}{228}& \frac{31}{456}\\ \frac{3}{38}& \frac{13}{228}& \frac{7}{456}\end{array}\right]$

Jeffrey Jordon