To determine the inverse of the matrix [tex]\( C = \left[\begin{array}{cc} 6 & -7 \\ -8 & 9 \end{array} \right] \)[/tex], we need to identify the matrix that, when multiplied with [tex]\( C \)[/tex], results in the identity matrix.
The matrix [tex]\( C \)[/tex] is a 2x2 matrix, and its inverse [tex]\( C^{-1} \)[/tex] can be represented as another 2x2 matrix:
[tex]\[ C^{-1} = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \][/tex]
When we multiply [tex]\( C \)[/tex] by [tex]\( C^{-1} \)[/tex], the product should be the identity matrix [tex]\( I \)[/tex]:
[tex]\[ C \cdot C^{-1} = I = \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right] \][/tex]
The given answer to the inverse of [tex]\( C \)[/tex] is:
[tex]\[ C^{-1} = \left[\begin{array}{cc} -4.5 & -3.5 \\ -4 & -3 \end{array}\right] \][/tex]
Let's verify this by checking if the multiplication of [tex]\( C \)[/tex] and [tex]\( C^{-1} \)[/tex] gives us the identity matrix.
Perform the matrix multiplication:
[tex]\[
C \cdot C^{-1} = \left[\begin{array}{cc} 6 & -7 \\ -8 & 9 \end{array}\right] \cdot \left[\begin{array}{cc} -4.5 & -3.5 \\ -4 & -3 \end{array}\right]
\][/tex]
Calculate the product:
[tex]\[
\begin{array}{cc}
(6 \cdot -4.5 + -7 \cdot -4) & (6 \cdot -3.5 + -7 \cdot -3) \\
(-8 \cdot -4.5 + 9 \cdot -4) & (-8 \cdot -3.5 + 9 \cdot -3)
\end{array}
\][/tex]
Simplify the entries:
[tex]\[
\begin{array}{cc}
(-27 + 28) & (-21 + 21) \\
(36 - 36) & (28 - 27)
\end{array}
=
\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]
\][/tex]
The result is indeed the identity matrix, confirming that the inverse provided is correct. Thus, the inverse of the matrix [tex]\( C \)[/tex] is:
[tex]\[ C^{-1} = \left[\begin{array}{cc} -4.5 & -3.5 \\ -4 & -3 \end{array}\right] \][/tex]
