91Ó°ÊÓ

The determinant of a matrix is a special number that can tell us a lot about the matrix itself. For a 2x2 matrix, the determinant is found using the formula: \[ \text{det}(A) = ad - bc \] Here, for the matrix \[ A = \begin{bmatrix} 1 & 1 \ -1 & 0 \end{bmatrix} \], we choose the elements a = 1, b = 1, c = -1, and d = 0. Putting these into the formula, we get: \[ \text{det}(A) = (1 \times 0) - (1 \times -1) = 0 + 1 = 1 \] A determinant of zero indicates that the matrix is not invertible. Fortunately, our determinant is 1, so the matrix is invertible!

The adjugate matrix, also known as the adjoint, is crucial for finding the inverse of a matrix. For a 2x2 matrix, the adjugate is formed by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. Let's consider our matrix again: \[ A = \begin{bmatrix} 1 & 1 \ -1 & 0 \end{bmatrix} \] To find the adjugate matrix:

• Swap the main diagonal elements, giving us \[ \begin{bmatrix} 0 & 1 \ -1 & 1 \end{bmatrix} \]
• Change the signs of the off-diagonal elements, resulting in: \[ \text{adj}(A) = \begin{bmatrix} 0 & 1 \ -1 & 1 \end{bmatrix} \]

Matrix inversion is the process of finding a matrix that, when multiplied by the original matrix, yields the identity matrix. For a matrix \( A \), its inverse \( A^{-1} \) is found using the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \] Given our matrix \( A = \begin{bmatrix} 1 & 1 \ -1 & 0 \end{bmatrix} \) and its determinant \( \text{det}(A) = 1 \), we use the adjugate matrix: \[ \text{adj}(A) = \begin{bmatrix} 0 & 1 \ -1 & 1 \end{bmatrix} \] Plug these into the formula: \[ A^{-1} = \frac{1}{1} \begin{bmatrix} 0 & 1 \ -1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -1 & 1 \end{bmatrix} \] So, the inverse of the original matrix is: \[ A^{-1} = \begin{bmatrix} 0 & 1 \ -1 & 1 \end{bmatrix} \]