To find the inverse of the matrix $A = \begin{bmatrix} 1 & 0 & -1 \\ -3 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix}$, we'll go through the following steps:

1. Check if the Matrix is Invertible:
   A matrix is invertible if its determinant is non-zero. So, let's calculate the determinant of $A$.

2. Calculate the Determinant: $\det(A) = 1 \cdot \begin{vmatrix} 1 & 3 \\ 0 & 1 \end{vmatrix} - 0 \cdot \begin{vmatrix} -3 & 3 \\ 0 & 1 \end{vmatrix} + (-1) \cdot \begin{vmatrix} -3 & 1 \\ 0 & 0 \end{vmatrix}$ Simplifying each minor: $= 1 \cdot (1 \cdot 1 - 3 \cdot 0) - 1 \cdot (-3 \cdot 0 - 1 \cdot 0) = 1$ Since the determinant is 1, $A$ is invertible.

3. Find the Inverse of $A$ Using Row Reduction or Adjoint Method: We can find $A^{-1}$ by using row reduction on the augmented matrix $[A | I]$ where $I$ is the identity matrix. Alternatively, using the cofactor method, the result will be: $A^{-1} = \begin{bmatrix} 1 & 0 & 1 \\ 3 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Thus, the inverse of $A$ is: $A^{-1} = \begin{bmatrix} 1 & 0 & 1 \\ 3 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Would you like further details on any of these steps?

Additional Questions

1. How can we verify that $A^{-1}$ is indeed the inverse of $A$?
2. What happens if the determinant of a matrix is zero?
3. How can we calculate the inverse of a 4x4 matrix?
4. Are there cases when a matrix doesn't have an inverse even if it's square?
5. How does row reduction work for finding the inverse of a matrix?

Tip

For small matrices (like 2x2 or 3x3), calculating the inverse manually is manageable, but for larger matrices, software or calculators are helpful to avoid errors.