91Ó°ÊÓ

The determinant is a special number that can be calculated from a square matrix. It helps determine if a matrix is invertible. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula: \[ \text{det}(A) = ad - bc \]. By plugging in the values, you can quickly find the determinant.

In our example with matrix \(\begin{bmatrix} 3 & 7 \ 8 & -2 \end{bmatrix}\), we apply the formula:

1. Multiply the diagonals: \(3 \times -2 = -6\)
2. Multiply the other diagonals: \(7 \times 8 = 56\)
3. Subtract the results: \(-6 - 56 = -62\)

Since the determinant is not zero (it's -62), our matrix is indeed invertible.

The adjugate (or adjoint) of a matrix is crucial in finding its inverse. For a 2x2 matrix, the adjugate is found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements.

To illustrate, let's find the adjugate of \(\begin{bmatrix} 3 & 7 \ 8 & -2 \end{bmatrix}\):
- Swap the main diagonal elements: \(\begin{bmatrix} -2 & 7 \ 8 & 3 \end{bmatrix}\)
- Change the signs of the off-diagonal elements: \(\begin{bmatrix} -2 & -7 \ -8 & 3 \end{bmatrix}\)

Thus, the adjugate matrix is \( \text{Adj}(A) = \begin{bmatrix} -2 & -7 \ -8 & 3 \end{bmatrix} \). Now we are ready to use it to find the inverse of our original matrix.

To find the inverse of a 2x2 matrix, we use its determinant and the adjugate matrix. The formula is:

\[ A^{-1} = \frac{1}{\text{det}(A)} \text{Adj}(A) \]

For our matrix \(\begin{bmatrix} 3 & 7 \ 8 & -2 \end{bmatrix}\), we know:
- \(\text{det}(A) = -62\)
- \(\text{Adj}(A) = \begin{bmatrix} -2 & -7 \ -8 & 3 \end{bmatrix}\)
Plug these into the formula:

1. Multiply every element of the adjugate matrix by \(\frac{1}{-62}\):
\( \begin{bmatrix} -2 & -7 \ -8 & 3 \end{bmatrix} \times \frac{1}{-62} = \begin{bmatrix} \frac{2}{62} & \frac{7}{62} \ \frac{8}{62} & \frac{-3}{62} \end{bmatrix} \)
2. Simplify the fractions:
\(\begin{bmatrix} \frac{2}{62} & \frac{7}{62} \ \frac{8}{62} & \frac{-3}{62} \end{bmatrix} = \begin{bmatrix} -\frac{1}{31} & -\frac{7}{62} \ -\frac{4}{31} & \frac{3}{62} \end{bmatrix}\)

Therefore, the inverse matrix is \( \begin{bmatrix} -\frac{1}{31} & -\frac{7}{62} \ -\frac{4}{31} & \frac{3}{62} \end{bmatrix} \). This completes the matrix inversion steps for a 2x2 matrix.