If the matrix A is such that `({:(1,3),(0,1):})A=({:(1,1),(0,-1):})`, then what is A equal to ?

Options

A. `{:[(1,0),(-3,1)]:}`
B. `{:[(1,-4),(0,1)]:}`
C. `{:[(1,-3),(0,1)]:}`
D. `{:[(1,-1),(-3,1)]:}`

Answer

To solve the equation \((1, 3; 0, 1)A = (1, 1; 0, -1)\) for the matrix \(A\), we will follow these steps:

### Step 1: Identify the matrices
Let \(B = (1, 3; 0, 1)\) and \(C = (1, 1; 0, -1)\). We need to find \(A\) such that \(BA = C\).

### Step 2: Find the inverse of matrix \(B\)
To isolate \(A\), we can multiply both sides of the equation by the inverse of \(B\):
\[ A = B^{-1}C \]

### Step 3: Calculate the determinant of \(B\)
The determinant of matrix \(B\) is calculated as follows:
\[ \text{det}(B) = (1)(1) - (3)(0) = 1 \]

### Step 4: Find the adjoint of matrix \(B\)
To find the adjoint of \(B\), we first calculate the cofactors:
- The cofactor \(B_{11} = 1\) (removing the first row and first column)
- The cofactor \(B_{12} = 0\) (removing the first row and second column)
- The cofactor \(B_{21} = -3\) (removing the second row and first column)
- The cofactor \(B_{22} = 1\) (removing the second row and second column)

Thus, the cofactor matrix is:
\[ \text{Cofactor}(B) = \begin{pmatrix} 1 & 0 \\ -3 & 1 \end{pmatrix} \]

The adjoint of \(B\) is the transpose of the cofactor matrix:
\[ \text{adj}(B) = \begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix} \]

### Step 5: Calculate the inverse of matrix \(B\)
Now we can find the inverse of \(B\):
\[ B^{-1} = \frac{\text{adj}(B)}{\text{det}(B)} = \begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix} \]

### Step 6: Multiply \(B^{-1}\) by \(C\) to find \(A\)
Now we compute \(A\):
\[ A = B^{-1}C = \begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix} \]

Calculating the product:
- First row, first column: \(1 \cdot 1 + (-3) \cdot 0 = 1\)
- First row, second column: \(1 \cdot 1 + (-3) \cdot (-1) = 1 + 3 = 4\)
- Second row, first column: \(0 \cdot 1 + 1 \cdot 0 = 0\)
- Second row, second column: \(0 \cdot 1 + 1 \cdot (-1) = 0 - 1 = -1\)

Thus, we have:
\[ A = \begin{pmatrix} 1 & 4 \\ 0 & -1 \end{pmatrix} \]

### Final Answer
The matrix \(A\) is:
\[ A = \begin{pmatrix} 1 & 4 \\ 0 & -1 \end{pmatrix} \]