Question

Find the inverse of [-6 , -3], [5, 3] if the inverse does not exist select "undefined"
write each matrix element in simplest form

Answer 1

To find the inverse of a 2x2 matrix, we use the formula for the inverse of a matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \):

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

In your case, the matrix is:

\[
\begin{pmatrix} -6 & -3 \\ 5 & 3 \end{pmatrix}
\]

Here, \( a = -6 \), \( b = -3 \), \( c = 5 \), and \( d = 3 \).

First, we calculate the determinant \( ad - bc \):

\[
ad = (-6)(3) = -18
\]
\[
bc = (-3)(5) = -15
\]
\[
ad - bc = -18 - (-15) = -18 + 15 = -3
\]

Since the determinant is not zero (\( -3 \neq 0 \)), the inverse exists. Now we can apply the formula for the inverse:

\[
A^{-1} = \frac{1}{-3} \begin{pmatrix} 3 & 3 \\ -5 & -6 \end{pmatrix}
\]

Multiplying each element by \(\frac{1}{-3}\):

\[
A^{-1} = \begin{pmatrix} \frac{3}{-3} & \frac{3}{-3} \\ \frac{-5}{-3} & \frac{-6}{-3} \end{pmatrix}
\]

This simplifies to:

\[
A^{-1} = \begin{pmatrix} -1 & -1 \\ \frac{5}{3} & 2 \end{pmatrix}
\]

Thus, the inverse of the matrix is:

\[
\begin{pmatrix} -1 & -1 \\ \frac{5}{3} & 2 \end{pmatrix}
\]