To determine the inverse of the matrix [tex]\(\begin{bmatrix} 2 & -1 \\ b & 4 \end{bmatrix}\)[/tex], we need to follow these steps:
1. Calculate the determinant of the matrix.
2. Check if the determinant is non-zero (since a matrix has an inverse if and only if its determinant is non-zero).
3. Find the adjugate (also called the adjoint) of the matrix.
4. Divide the adjugate by the determinant to get the inverse of the matrix.
Let's carry out these steps in detail.
### Step 1: Calculate the Determinant
The determinant [tex]\(\text{det}(A)\)[/tex] of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\)[/tex] is found by the formula:
[tex]\[
\text{det}(A) = ad - bc
\][/tex]
For our matrix [tex]\(\begin{bmatrix} 2 & -1 \\ b & 4 \end{bmatrix}\)[/tex]:
[tex]\[
\text{det}(A) = (2)(4) - (-1)(b) = 8 + b = 8 + b
\][/tex]
### Step 2: Check if the Determinant is Non-zero
We need to ensure that [tex]\(8 + b \neq 0\)[/tex]. Assuming [tex]\(b \neq -8\)[/tex], the determinant is non-zero and we can proceed.
### Step 3: Find the Adjugate Matrix
The adjugate matrix of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\)[/tex] is:
[tex]\[
\text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\][/tex]
For our matrix [tex]\(\begin{bmatrix} 2 & -1 \\ b & 4 \end{bmatrix}\)[/tex], the adjugate matrix is:
[tex]\[
\text{adj}(A) = \begin{bmatrix} 4 & 1 \\ -b & 2 \end{bmatrix}
\][/tex]
### Step 4: Calculate the Inverse Matrix
The inverse [tex]\(A^{-1}\)[/tex] of a matrix [tex]\(A\)[/tex] is given by:
[tex]\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\][/tex]
Using the results obtained:
[tex]\[
\text{det}(A) = 8 + b
\][/tex]
[tex]\[
\text{adj}(A) = \begin{bmatrix} 4 & 1 \\ -b & 2 \end{bmatrix}
\][/tex]
The inverse matrix is:
[tex]\[
A^{-1} = \frac{1}{8 + b} \cdot \begin{bmatrix} 4 & 1 \\ -b & 2 \end{bmatrix} = \begin{bmatrix} \frac{4}{8 + b} & \frac{1}{8 + b} \\ \frac{-b}{8 + b} & \frac{2}{8 + b} \end{bmatrix}
\][/tex]
### Conclusion
The correct answer for the inverse of the matrix [tex]\(\begin{bmatrix} 2 & -1 \\ b & 4 \end{bmatrix}\)[/tex] is:
[tex]\[
(1) \begin{bmatrix} \frac{4}{8 + b} & \frac{1}{8 + b} \\ \frac{-b}{8 + b} & \frac{2}{8 + b} \end{bmatrix}
\][/tex]
