The image contains a matrix-related problem in Gujarati. Here’s a breakdown of the given problem in English:

1. Matrix Definition: Matrix $A$ is defined as: $A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & 1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$

2. Objective:

   • Find the inverse of matrix $A$, denoted $A^{-1}$, if it exists.
   • Use the inverse of matrix $A$ to solve the following system of equations: $2x - y - z = 8$ $-2y + z = 7$ $x - 2y = 10$

Solution Outline

1. Step 1: Calculate $A^{-1}$.

   • To find $A^{-1}$, we will compute the determinant of $A$ and use the adjugate (or adjoint) matrix method or row reduction.

2. Step 2: Solve the System Using $A^{-1}$.

   • Rewrite the system of equations in matrix form as $AX = B$, where $B = \begin{bmatrix} 8 \\ 7 \\ 10 \end{bmatrix}$.
   • Calculate $X = A^{-1}B$ to find the values of $x$, $y$, and $z$.

Would you like me to perform the calculations for $A^{-1}$ and solve for $x$, $y$, and $z$?

Here are related questions that expand on this topic:

1. What is the process of calculating the determinant of a 3x3 matrix?
2. How can we check if a matrix is invertible?
3. How is matrix multiplication applied when solving linear systems?
4. What are alternative methods to solve a system of equations, such as Gaussian elimination?
5. How does the adjugate matrix method work in finding the inverse?

Tip: Remember that not all matrices have inverses; a matrix must be non-singular (determinant not equal to zero) to have an inverse.