Find the inverse of the matrix $\left[ \begin{array} { c c } 6 & 12 \ - 18 & - 30 \end{array} \right]$.
A) $\frac { 1 } { 7 } \left[ \begin{array} { c c } 6 & 12 \ - 18 & - 30 \end{array} \right]$
B) $\frac { 1 } { 6 } \left[ \begin{array} { c c } - 5 & - 2 \ 3 & 1 \end{array} \right]$
C) $\left[ \begin{array} { c c } - 6 & - 12 \ 30 & 18 \end{array} \right]$
D) $\left[ \begin{array} { c c } 6 & - 18 \ 24 & - 42 \end{array} \right]$
E) $\frac { 1 } { 13 } \left[ \begin{array} { c c } - 6 & - 12 \ - 18 & 30 \end{array} \right]$

Question

Quizplus · Accepted Answer

To find the inverse of a $2	imes 2$ matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, we first calculate the determinant: $$\det\begin{pmatrix} a & b \ c & d \end{pmatrix} = ad-bc.$$If the determinant is nonzero, then the inverse is given by: $$\begin{pmatrix} a & b \ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}.$$For the given matrix, the determinant is $6\cdot(-30)-12\cdot(-18)=-360
eq 0$, so the inverse is given by: $$\frac{1}{-360}\begin{pmatrix} -30 & -12 \ 18 & 6 \end{pmatrix} = \frac16\begin{pmatrix} 5 & 2 \ -3 & -1 \end{pmatrix}.$$Thus, the answer is $\boxed{	extbf{(B)}}$.

Find the Inverse of the Matrix [612−18−30]\left[ \begin{array} { c c } 6 & 12 \\ - 18 & - 30 \end{array} \right][6−18​12−30​]

Find the Inverse of the Matrix $\left[ \begin{array} { c c } 6 & 12 \\ - 18 & - 30 \end{array} \right]$