Find the inverse of the matrix $\left[ \begin{array} { c c } 8 & 16 \ - 24 & - 40 \end{array} \right]$.
A) $\frac { 1 } { 7 } \left[ \begin{array} { c c } 8 & 16 \ - 24 & - 40 \end{array} \right]$
B) $\frac { 1 } { 8 } \left[ \begin{array} { c c } - 5 & - 2 \ 3 & 1 \end{array} \right]$
C) $\left[ \begin{array} { c c } - 8 & - 16 \ 40 & 24 \end{array} \right]$
D) $\left[ \begin{array} { c c } 8 & - 24 \ 32 & - 56 \end{array} \right]$
E) $\frac { 1 } { 13 } \left[ \begin{array} { c c } - 8 & - 16 \ - 24 & 40 \end{array} \right]$

Question

Quizplus · Accepted Answer

To find the inverse of a matrix, we can use the following formula:

$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$

where $a,b,c,d$ are the entries of the matrix, and $\det(A) = ad - bc$ is the determinant of the matrix.

Using this formula with the given matrix, we have:

$\det \begin{bmatrix} 8 & 16 \ -24 & -40 \end{bmatrix} = (8)(-40) - (16)(-24) = -320 - (-384) = 64$

$A^{-1} = \frac{1}{64} \begin{bmatrix} -40 & -16 \ 24 & 8 \end{bmatrix} = \begin{bmatrix} -5/8 & -1/4 \ 3/8 & 1/8 \end{bmatrix}$

Therefore, the answer is choice B.

Find the Inverse of the Matrix [816−24−40]\left[ \begin{array} { c c } 8 & 16 \\ - 24 & - 40 \end{array} \right][8−24​16−40​]

Find the Inverse of the Matrix $\left[ \begin{array} { c c } 8 & 16 \\ - 24 & - 40 \end{array} \right]$