Q69. - Algebra 2

Question

Find the inverse of the matrix $\left[\begin{array}{c}\begin{array}{cc}4& 3\\ 2& 1\end{array}\end{array}\right]$  if it exists.

Step-by-Step Solution

Answer

The inverse of the given matrix is $\frac{1}{2}\left[\begin{array}{cc}-1& 3\\ 2& -4\end{array}\right]$ .

Step 1. Write down the given matrix.

The given matrix is say, $P=\left[\begin{array}{cc}4& 3\\ 2& 1\end{array}\right]$ .

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:

$\begin{array}{c}P=\left[\begin{array}{cc}4& 3\\ 2& 1\end{array}\right]\mathrm{....}\left(\text{Given}\right)\\ \left|P\right|=4-6\\ =-2\end{array}$

Now evaluating inverse of  $P=\left[\begin{array}{cc}4& 3\\ 2& 1\end{array}\right]$ by the formula stated above.

$\begin{array}{c}{P}^{-1}=\frac{1}{-2}\left[\begin{array}{cc}1& -3\\ -2& 4\end{array}\right]\\ =\frac{1}{2}\left[\begin{array}{cc}-1& 3\\ 2& -4\end{array}\right]\end{array}$

Step 4. Conclusion.

Hence the inverse of the given matrix is$\frac{1}{2}\left[\begin{array}{cc}-1& 3\\ 2& -4\end{array}\right]$  .