The inverse of the matrix `{:[(1,3),(3,10)]:}` is equal to

Options

A. `{:[(10,3),(3,1)]:}`
B. `{:[(10,-3),(-3,1)]:}`
C. `{:[(1,3),(3,10)]:}`
D. `{:[(-1,-3),(-3,-10)]:}`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 3 \\ 3 & 10 \end{pmatrix} \), we can follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 1 \)
- \( b = 3 \)
- \( c = 3 \)
- \( d = 10 \)

Calculating the determinant:
\[ \text{det}(A) = (1)(10) - (3)(3) = 10 - 9 = 1 \]

### Step 2: Find the Adjoint of Matrix A
The adjoint of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \( A \):
- \( d = 10 \)
- \( -b = -3 \)
- \( -c = -3 \)
- \( a = 1 \)

Thus, the adjoint of \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} 10 & -3 \\ -3 & 1 \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of a matrix \( A \) can be calculated using the formula:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{1} \begin{pmatrix} 10 & -3 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} 10 & -3 \\ -3 & 1 \end{pmatrix} \]

### Final Answer
The inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} 10 & -3 \\ -3 & 1 \end{pmatrix} \]

---