Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c }
d & - b \
- c & a
\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists). $$\left[ \begin{array} { c c }
6 & 7 \
- 5 & 13
\end{array} \right]$$
A) $$\frac { 1 } { 113 } \left[ \begin{array} { c c }
13 & - 7 \
5 & 6
\end{array} \right]$$
B) $$\frac { 1 } { 113 } \left[ \begin{array} { c c }
- 13 & 7 \
5 & - 6
\end{array} \right]$$
C) $$\frac { 1 } { 113 } \left[ \begin{array} { c c }
- 13 & - 7 \
5 & 6
\end{array} \right]$$
D) $$\frac { 1 } { 113 } \left[ \begin{array} { c c }
13 & 7 \
5 & 6
\end{array} \right]$$
E) $$\frac { 1 } { 113 } \left[ \begin{array} { c c }
- 13 & - 7 \
- 5 & - 6
\end{array} \right]$$

Question

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists).​ $$\left[ \begin{array} { c c } 
6 & 7 \
- 5 & 13
\end{array} \right]$$ ​
A)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & - 7 \
5 & 6
\end{array} \right]$$
B)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & 7 \
5 & - 6
\end{array} \right]$$
C)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
5 & 6
\end{array} \right]$$
D)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
13 & 7 \
5 & 6
\end{array} \right]$$
E)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 
- 13 & - 7 \
- 5 & - 6
\end{array} \right]$$

Quizplus · Accepted Answer

The determinant of the matrix is $ad - bc = (6)(13) - (7)(-5) = 78 + 35 = 113$. According to the inverse formula, the inverse matrix is $\frac{1}{113}\left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right]$ which, when applied to the given matrix, results in $\frac{1}{113}\left[ \begin{array}{cc} 13 & -7 \ 5 & 6 \end{array} \right]$.

Use the Inverse Formula A−1=1ad−bc[d−b−ca]A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]A−1=ad−bc1​[d−c​−ba​]

Use the Inverse Formula $A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$