To determine the inverse of the matrix
[tex]\[ A = \begin{bmatrix} 1 & 0 \\ -6 & 4 \end{bmatrix}, \][/tex]
we need to follow the standard steps for finding the inverse of a 2x2 matrix.
The inverse of a 2x2 matrix
[tex]\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \][/tex]
is given by
[tex]\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}, \][/tex]
provided that [tex]\( ad - bc \neq 0 \)[/tex].
Let's apply this formula to our matrix [tex]\( A \)[/tex].
1. Identify the elements:
[tex]\[ a = 1, \quad b = 0, \quad c = -6, \quad d = 4. \][/tex]
2. Calculate the determinant [tex]\( ad - bc \)[/tex]:
[tex]\[ ad - bc = (1 \cdot 4) - (0 \cdot -6) = 4 - 0 = 4. \][/tex]
Since the determinant is not zero, the matrix is invertible.
3. Apply the formula for the inverse:
[tex]\[ A^{-1} = \frac{1}{4} \begin{bmatrix} 4 & 0 \\ 6 & 1 \end{bmatrix}. \][/tex]
4. Simplify the matrix:
[tex]\[ A^{-1} = \frac{1}{4} \begin{bmatrix} 4 & 0 \\ 6 & 1 \end{bmatrix} = \begin{bmatrix} 4/4 & 0 \\ 6/4 & 1/4 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1.5 & 0.25 \end{bmatrix}. \][/tex]
Thus, the inverse of the matrix [tex]\( A \)[/tex] is:
[tex]\[ A^{-1} = \begin{bmatrix} 1 & 0 \\ 1.5 & 0.25 \end{bmatrix}. \][/tex]
