Two matrices X Y MnR are said to anticommute if XY Y X Let

Two matrices X, Y ? Mn(R) are said to anti-commute if XY = ?Y X. Let X1, X2, . . . Xk ? Mn(R) anti-commute (i.e., XiXj = ?XjXi for all i and j) and suppose that X2 i 6= 0n for all i. Show that the matrices X1, X2, . . . Xk are linearly independent..

Two matrices X, Y € M, (R) are said to anti-commute if XY =-YX. Let Xi, X2, . . . X. E M(R) ani-commute (i.e., X,X Mn(R) anti-commute (i.e., Xix, =-X,X, for all i and J) and suppose that X?On n(R) anti-commute (1.e., Ai for all i. Show that the matrices Xi,X2, Xk are linearly independent.

Solution

suppose there exist scalars c1,c2,c3.......ck

c1y1+c2y2+......+ckyk=0

To show x1,x2,.....xk are linearly dependent we have to show that

c1=c2=c3=.......=ck=0

Given xy=-yx

taking y from this to substitute in above equation gives xy +yx=0

which can be written as y=-2x

substituting this

c1 (-2x1) +c2(-2x2) +........ + ck (-2x2) =0

==> 2(C1x1+C2x2+...........+CkXk) = 0

multiplying both sides by2^-1 yields

2^-1 [ 2(C1x1+C2x2+...........+CkXk) ]= 2^-1*0

2^-1 * 2 is 1

which can be written as

C1x1+C2x2+...........+CkXk=0

This shows x1,x2,x3.....xk are linearly dependent.