Question

Find the inverse of each matrix, if it exists.

$\left[\begin{array}{cc}3& 1\\ -4& 1\end{array}\right]$

Short answer

The inverse of the given matrix is$\left[\begin{array}{cc}\frac{1}{7}& -\frac{1}{7}\\ \frac{4}{7}& \frac{3}{7}\end{array}\right]$.

Step-by-step solution

Step 1. Given Information.

Given, to find the inverse of the matrix $\left[\begin{array}{cc}3& 1\\ -4& 1\end{array}\right]$.

Step 2. Calculation.

The inverse of matrix$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$,where$ad-bc\ne 0$:

Here$a=3,b=1,c=-4,d=1$.

Checking if the inverse exists:

$\begin{array}{l}ad-bc=\left(3\right)\left(1\right)-\left(1\right)\left(-4\right)\\ ad-bc=3+4\\ ad-bc=7\end{array}$

Hence, the inverse exists.

The inverse of the given matrix is:

$\begin{array}{l}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ A^{-1}=\frac{1}{\left(3\right)\left(1\right)-\left(1\right)\left(-4\right)}\left[\begin{array}{cc}1& -1\\ -\left(-4\right)& 3\end{array}\right]\\ A^{-1}=\frac{1}{3+4}\left[\begin{array}{cc}1& -1\\ 4& 3\end{array}\right]\\ A^{-1}=\frac{1}{7}\left[\begin{array}{cc}1& -1\\ 4& 3\end{array}\right]\\ A^{-1}=\left[\begin{array}{cc}\frac{1}{7}& -\frac{1}{7}\\ \frac{4}{7}& \frac{3}{7}\end{array}\right]\end{array}$

Step 3. Conclusion.

Hence, the inverse of the given matrix is $\left[\begin{array}{cc}\frac{1}{7}& -\frac{1}{7}\\ \frac{4}{7}& \frac{3}{7}\end{array}\right]$.