Let *A* = (*a*_{i,j} be an *n × n* matrix and *x* = (*x*_{1}, *x*_{2},…, *x _{n}*),

If det(*A*) ≠ 0, then for *every y*, the linear system of Equations (3.20) has a unique solution *x*,

Gaussian elimination is a process in which transformations are applied to an invertible matrix *A* to produce the identity matrix *I* and thereby obtain the solution for *x* in Equation (3.20).

*R*_{r,s}(*v*) (*r ≠ s*) is the*n × n*matrix equal to the identity matrix, except that the element in position (*r*,*s*) of*R*_{r,s}(*v*) is*v*. For example when*n*= 4If

then

*Pre*multiplication of*A*by*R*_{r, s}(*v*) replaces the*r*th row of*A*by the sum of*v*times the*s*th row of*A*and- The
*r*th row of*A*.The inverse of

*R*_{r, s}(*v*) is*R*_{r, s}(−*v*).

*C*_{r, s}(*v*) (*r ≠ s*) is the*n × n*matrix, which is equal to the identity matrix except that the element in position (*r*,*s*) of*C*_{r,s}(*v*) is*v*. For example, when*n*= 4If

then

*Post*multiplication ...

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