By Magar E. Mager

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A I n — I nA (2-17) in addition each matrix commutes with itself A A = AA. N o w if we 2 3 2 4 3 n n _ 1 define A = AA, A = A A , A = A A , . . , A = A A , it is easy to n m w n n+m show, by induction, that A A = A A = A . If for a matrix A we +1 have A * = A, where k is a positive integer, t h e n A is said to be nil2 potent with index k. If in particular k = 1 so that A = A, t h e n A is called indempotent. 25 5. 2 TRANSPOSITION T h e transpose of a matrix is defined as t h e matrix obtained by interchanging t h e rows a n d columns of t h e original matrix.

A system of η n o n homogenous equations in η u n k n o w n s has a u n i q u e solution provided the rank of the coefficient matrix is w, that is if A is a nonsingular matrix. If we have a system of η nonhomogenous equations in η u n k n o w n written A x = b, we can solve this system by means of Cramer^ rule. T h i s is done by defining A^ for i = 1, 2, . . , η as the matrix obtained from A by replacing its ith column by the column vector b. T h e u n i q u e solution for the n o n h o m o g e n o u s equations is x2 = I A 2 1 / 1 A | , .

F o r example for the t e r m — 8xxx2 we can choose the corresponding elements of t h e matrix A as a12 = 0 and a21= — 8 or we can have a12 = 2 and a21 = —10 or we may have a21 = a12 = — 4. Since we can always choose a{j = a^ for all ij t h e n it follows that the matrix of our quadratic form above is given by the symmetric matrix 2 4 8 -4 4 0 8i 0 -14. (2-63) I n particular this can be done for all quadratic forms. W e shall consider only symmetric matrices in what follows. T h e rank of the quadratic form is defined as the rank of A.