The inverse of matrix `[[2, 1], [7, 4]]` is

Options

A. `-[[4, 1], [7, 2]]`
B. `[[4, 1], [7, 2]]`
C. `-[[4, -1], [-7, 2]]`
D. `[[4, -1], [-7, 2]]`

Answer

To find the inverse of the matrix \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\), we will follow these steps:

### Step 1: Calculate the Determinant of the Matrix
The determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \(A = \begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\):
- \(a = 2\)
- \(b = 1\)
- \(c = 7\)
- \(d = 4\)

Calculating the determinant:
\[ \text{det}(A) = (2)(4) - (1)(7) = 8 - 7 = 1 \]

### Step 2: Calculate the Cofactor Matrix
The cofactor matrix is obtained by applying the sign convention and calculating the minors of each element. The sign convention for a \(2 \times 2\) matrix is:
\[ \begin{bmatrix} + & - \\ - & + \end{bmatrix} \]
Calculating the cofactors:
- For element \(2\) (position (1,1)), the minor is \(4\) (since we remove the first row and first column).
- For element \(1\) (position (1,2)), the minor is \(7\) (remove first row and second column, sign is negative).
- For element \(7\) (position (2,1)), the minor is \(1\) (remove second row and first column, sign is negative).
- For element \(4\) (position (2,2)), the minor is \(2\) (remove second row and second column).

Thus, the cofactor matrix \(C(A)\) is:
\[ C(A) = \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix} \]

### Step 3: Calculate the Adjoint of the Matrix
The adjoint of a matrix is the transpose of the cofactor matrix. So we need to transpose \(C(A)\):
\[ \text{adj}(A) = C(A)^T = \begin{bmatrix} 4 & -7 \\ -1 & 2 \end{bmatrix} \]

### Step 4: Calculate the Inverse of the Matrix
Using the formula for the inverse of a matrix:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{1} \cdot \begin{bmatrix} 4 & -7 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 4 & -7 \\ -1 & 2 \end{bmatrix} \]

### Final Answer
The inverse of the matrix \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\) is:
\[ \begin{bmatrix} 4 & -7 \\ -1 & 2 \end{bmatrix} \]

---