Find X, if `A+X=I`, where
`A=[(1,4,-1),(3,4,7),(5,1,6)]`

Answer

To solve the equation \( A + X = I \), where \( A = \begin{pmatrix} 1 & 4 & -1 \\ 3 & 4 & 7 \\ 5 & 1 & 6 \end{pmatrix} \) and \( I \) is the identity matrix, we can follow these steps:

### Step 1: Write down the identity matrix \( I \)
The identity matrix \( I \) for a 3x3 matrix is given by:
\[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Rearrange the equation to solve for \( X \)
From the equation \( A + X = I \), we can rearrange it to find \( X \):
\[ X = I - A \]

### Step 3: Calculate \( I - A \)
Now we will subtract matrix \( A \) from the identity matrix \( I \):
\[ X = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} - \begin{pmatrix} 1 & 4 & -1 \\ 3 & 4 & 7 \\ 5 & 1 & 6 \end{pmatrix} \]

### Step 4: Perform the subtraction element-wise
We will subtract the corresponding elements of the matrices:
\[ X = \begin{pmatrix} 1 - 1 & 0 - 4 & 0 - (-1) \\ 0 - 3 & 1 - 4 & 0 - 7 \\ 0 - 5 & 0 - 1 & 1 - 6 \end{pmatrix} \]
Calculating each element:
- First row: \( 1 - 1 = 0 \), \( 0 - 4 = -4 \), \( 0 - (-1) = 1 \)
- Second row: \( 0 - 3 = -3 \), \( 1 - 4 = -3 \), \( 0 - 7 = -7 \)
- Third row: \( 0 - 5 = -5 \), \( 0 - 1 = -1 \), \( 1 - 6 = -5 \)

Thus, we have:
\[ X = \begin{pmatrix} 0 & -4 & 1 \\ -3 & -3 & -7 \\ -5 & -1 & -5 \end{pmatrix} \]

### Final Answer
The matrix \( X \) is:
\[ X = \begin{pmatrix} 0 & -4 & 1 \\ -3 & -3 & -7 \\ -5 & -1 & -5 \end{pmatrix} \]