Multiplicative inverse of the matrix `[[2,1],[7,4]]` is (i) `[[4,-1],[-7,-2]]` (ii) `[[-4,-1],[7,-2]]` (iii) `[[4,-1],[7,2]]` (iv) `[[4,-1],[-7,2]]`

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

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = 1 \)
- \( c = 7 \)
- \( d = 4 \)

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

### Step 2: Find the Cofactor Matrix
The cofactor matrix is found by calculating the cofactors of each element in the matrix. The cofactor \( C_{ij} \) of an element \( a_{ij} \) is given by:
\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]
where \( M_{ij} \) is the determinant of the matrix formed by deleting the \( i \)-th row and \( j \)-th column.

Calculating the cofactors:
- \( C_{11} = 4 \) (from element 2)
- \( C_{12} = -7 \) (from element 1)
- \( C_{21} = -1 \) (from element 7)
- \( C_{22} = 2 \) (from element 4)

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

### Step 3: Find the Adjoint of Matrix A
The adjoint of a matrix is the transpose of the cofactor matrix. Therefore, we take the transpose:
\[ \text{Adj}(A) = \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix} \]

### Step 4: Calculate the Inverse of Matrix A
The multiplicative inverse \( A^{-1} \) is given by:
\[ A^{-1} = \frac{\text{Adj}(A)}{\text{det}(A)} \]
Since the determinant is 1, we have:
\[ A^{-1} = \text{Adj}(A) = \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix} \]

### Conclusion
The multiplicative inverse of the matrix \( A = \begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix} \) is:
\[ A^{-1} = \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix} \]

### Answer
The correct option is (iv) \( \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix} \).

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