91Ó°ÊÓ

To find the inverse of a matrix, the first key step is computing the determinant. The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible.
The formula for the determinant of a 2x2 matrix \(A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right]\) is given by:
\[ \text{det}(A) = ad - bc \]
In our exercise, we have:
\[ A = \left[\begin{array}{rr} 2 & 1 \ -4 & 7 \end{array}\right] \]
So, the determinant is:
\[ \text{det}(A) = (2 \times 7) - (1 \times -4) = 14 + 4 = 18 \]
This tells us that the matrix is invertible, because its determinant is not zero.

The adjugate matrix plays an important role in finding the inverse of a matrix. It is constructed by performing specific operations on the original matrix.
For a 2x2 matrix \(A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right]\), the adjugate (or adjoint) matrix is found by:
1. Swapping the elements on the main diagonal.
2. Changing the signs of the off-diagonal elements.
Thus, for our example matrix:
\[ A = \left[\begin{array}{rr} 2 & 1 \ -4 & 7 \end{array}\right] \]
The adjugate matrix is:
\[ \text{Adj}(A) = \left[\begin{array}{rr} 7 & -1 \ 4 & 2 \end{array}\right] \]
Notice how the elements 2 and 7 are swapped, and the signs of 1 and -4 are changed to -1 and 4, respectively.

The goal of matrix inversion is to find a matrix that, when multiplied with the original matrix, results in the identity matrix. For a 2x2 matrix \(A\), if we have calculated the determinant (det(A)) and the adjugate matrix (Adj(A)), the inverse matrix \(A^{-1}\) is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \text{Adj}(A) \]
In our given solution, since \( \text{det}(A) = 18 \) and:
\[ \text{Adj}(A) = \left[\begin{array}{rr} 7 & -1 \ 4 & 2 \end{array}\right] \]
The inverse matrix is:
\[ A^{-1} = \frac{1}{18} \left[\begin{array}{rr} 7 & -1 \ 4 & 2 \end{array}\right] \]
By multiplying each element of the adjugate matrix by \( \frac{1}{18} \), we get:
\[ A^{-1} = \left[\begin{array}{rr} \frac{7}{18} & -\frac{1}{18} \ \frac{2}{9} & \frac{1}{9} \end{array}\right] \]
Finally, simplify each fraction where possible to keep the matrix in its simplest form.