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Determine the inverse of the matrix [tex]C=\left[\begin{array}{cc}6 & -7 \\ -8 & 9\end{array}\right][/tex].

A. [tex]C^{-1}=\left[\begin{array}{ll}9 & 7 \\ 8 & 6\end{array}\right][/tex]

B. [tex]C^{-1}=\left[\begin{array}{cc}-9 & 7 \\ 8 & -6\end{array}\right][/tex]

C. [tex]C^{-1}=\left[\begin{array}{cc}-4.5 & -3.5 \\ -4 & -3\end{array}\right][/tex]

D. [tex]C^{-1}=\left[\begin{array}{cc}4.5 & 3.5 \\ 4 & 3\end{array}\right][/tex]


Sagot :

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]