Question: [Finding solution of system of simultaneous linear equations]

Consider the following system of linear equations with three unknowns.

6x+3y+x = 22,

x+4y−2z = 12,

4x−y+5z = 10

1. Write the equations in matrix notation
   Av = b,
   where A is the 3×3 matrix containing the coefficients, v is the 3×1 column vector of the unknowns and b is 3×1 column vector containing terms on the right hand side of the three equations.
2. It is given that the matrix A is invertible and its inverse is as follows.
   \[{{A}^{-1}}=\frac{1}{52}\left[ \begin{matrix} 18 & -16 & -10 \\ -13 & 26 & 13 \\ -17 & 18 & 21 \\ \end{matrix} \right]\]
   Use the inverse matrix to solve for the values of the unknowns.
3. Find the determinant of the matrix A (which must be non-zero, Why?).
4. Apply Cramer’s rule to solve for the values of the unknowns.

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