Matrix Equations and Systems of Linear Equations

A system of linear equations having the same number of variables as the number of equations
may be solved by rewriting the system as a matrix equation and then solving the matrix equation
using the identity matrix and the inverse matrix

If the system either has
     fewer variables than equations
or
     more variables than equations
then we must go back to the Gauss-Jordan method of elimination

The formal process of solving a matrix equation is much like the process of solving equations
involving real numbers, except that
     matrix multiplication is not commutative
and
     a nonzero matrices may have no inverse

Basic Properties of matrices
Assume that all products and sums are defined for the matrices A, B, C, I, and 0.

Addition Properties
              Associative:      (A + B) + C = A + (B + C)
            Commutative:      A + B = B + A
     Additive Identity:      A + 0 = 0 + A = A
     Additive Inverse:      A + (-A) = (-A) + A = 0

Multiplication Properties
                       Associative:      A(BC) = (AB)C
                     Commutative:      AB = BA
     Multiplicative Identity:      AI = IA = A
     Multiplicative Inverse:      If A is a square matrix and A^-1 exists,
                                                   then  A A^-1 = A^-1A = I

Combined Properties
        Left Distributive:      A(B + C) = AB + AC
     Right Distributive:      (B + C)A = BA + CA

Equality
                         Addition:      If A = B, then A + C = B + C
        Left Multiplication:      If A = B, then CA = CB
     Right Multiplication:      If A = B, then AC = BC




Matrix Equations
Solving a Matrix Equation – see Example 1, pages 757 – 758, of the textbook


Matrix Equations and Systems of Linear Equations
Independent systems of linear equations with the same number of variables as equations
may be solved by first converting the system into a matrix equation of the form
                                                                      AX = B
If A has an inverse A^-1, then
                                                                      A^-1(AX)= A^-1B
or
                                                                      A^-1AX = A^-1B
Since
                                                                      A^-1A = I
we get the solution
                                                                                X = A^-1B

If the number of equations in a system is equal to the number of variables and the coefficient matrix
has an inverse, then the system has a unique solution that can be computed by using the inverse of
the coefficient matrix to solve the corresponding matrix equation


     See Examples 2 – 4, pages 758 – 763, of the textbook


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