Q85. - Algebra 2

Question

Find the inverse of the matrix

$\left[\begin{array}{cc}-5& 6\\ -11& 3\\ & \end{array}\right]$

 if it exists.

Step-by-Step Solution

Answer

The inverse of the given matrix is $\frac{1}{\left|51\right|}\left[\begin{array}{cc}3& -6\\ 11& -5\end{array}\right]$

Step 1. Write down the given equation.

The given matrix is say

$P=\left[\begin{array}{cc}-5& 6\\ -11& 3\\ & \end{array}\right]$

.

Step 2. Concept used.

The inverse of the matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

 exits if $\left|A\right|\ne 0$. And inverse of $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

 is given by the formula:

$A^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ where $\left|A\right|\ne 0$

Step 3. Evaluating inverse of given matrix.

The inverse of the matrix is evaluated as:

$\begin{array}{l}P=\left[\begin{array}{cc}-5& 6\\ -11& 3\end{array}\right]\mathrm{....}\left(\text{Given}\right)\\ \left|P\right|=-15+66\\ =51\end{array}$

Now evaluating inverse of 

$P=\left[\begin{array}{cc}-5& 6\\ -11& 3\end{array}\right]$

by the formula stated above.

$P^{-1}=\frac{1}{\left|51\right|}\left[\begin{array}{cc}3& -6\\ 11& -5\end{array}\right]$

Step 4. Conclusion.

Hence the inverse of the given matrix is 

$\frac{1}{\left|51\right|}\left[\begin{array}{cc}3& -6\\ 11& -5\end{array}\right]$