If `[[1, 1], [0, 1]]A=I`, then A=

Options

A. `[[1, 1], [0, -1]]`
B. `[[-1, 1], [0, 1]]`
C. `[[1, 1], [0, 1]]`
D. `[[1, -1], [0, 1]]`

Answer

To solve the equation \(\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} A = I\), where \(I\) is the identity matrix, we need to find the matrix \(A\).

### Step 1: Identify the matrices involved
Let \(B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) and \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).

### Step 2: Rewrite the equation
The equation can be rewritten as:
\[ B A = I \]

### Step 3: Multiply both sides by \(B^{-1}\)
To isolate \(A\), we will multiply both sides of the equation by the inverse of \(B\) (denoted as \(B^{-1}\)):
\[ B^{-1} B A = B^{-1} I \]
This simplifies to:
\[ I A = B^{-1} \]
Thus, we have:
\[ A = B^{-1} \]

### Step 4: Calculate the inverse of matrix \(B\)
To find \(B^{-1}\), we will use the formula for the inverse of a \(2 \times 2\) matrix:
\[ \text{If } B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ then } B^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \(B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\):
- \(a = 1\), \(b = 1\), \(c = 0\), \(d = 1\)

### Step 5: Calculate the determinant of \(B\)
The determinant \(det(B) = ad - bc = (1)(1) - (1)(0) = 1\).

### Step 6: Apply the inverse formula
Now, substituting into the inverse formula:
\[ B^{-1} = \frac{1}{1} \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \]

### Step 7: State the final result
Thus, we find:
\[ A = B^{-1} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \]

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