To find the inverse of the matrix $A$ given by:

$A = \begin{bmatrix} -1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{bmatrix}$

we need to compute $A^{-1}$, which requires calculating the matrix's determinant and its adjugate.

Step-by-Step Solution:

1. Calculate the Determinant of $A$: $\det(A) = -1 \begin{vmatrix} -2 & 1 \\ 1 & -2 \end{vmatrix} - 1 \begin{vmatrix} 1 & 1 \\ 1 & -2 \end{vmatrix} + 1 \begin{vmatrix} 1 & -2 \\ 1 & 1 \end{vmatrix}$ Each minor determinant can be calculated separately.

2. Find the Matrix of Cofactors by determining the cofactors of each element in $A$.

3. Transpose the Matrix of Cofactors to get the adjugate matrix.

4. Multiply the Adjugate by $\frac{1}{\det(A)}$ to get $A^{-1}$.

Let me know if you would like a detailed walkthrough of each step, or if you want to calculate a specific part together.

Would you like a detailed breakdown, or do you have any questions?

Related Questions:

1. How do you calculate the determinant of a 3x3 matrix?
2. What is the formula for the adjugate of a matrix?
3. Why do we transpose the cofactor matrix to find the adjugate?
4. Can every square matrix be inverted?
5. How can we verify if a calculated inverse matrix is correct?

Tip:

When calculating determinants, simplifying each minor step-by-step can help avoid common arithmetic errors.