Let A E Mnxn(F) and B = aA, with a E F. Prove by induction that det(B) = a" det(A), for all positive integer n.

Let x and y be in Z, not both zero, then x2+y2Z+.

Let be as described in the proof of Theorem. Give a specific example of a positive element of .

31. Prove statement of Theorem : for all integers and .

Assume the statement from Exercise 30 in section 2.1 that for all and in . Use this assumption and mathematical induction to prove that for all positive integers and arbitrary integers .

Prove that addition is associative in Q.

Use the second principle of Finite Induction to prove that every positive integer n can be expressed in the form n=c0+c13+c232+...+cj13j1+cj3j, where j is a nonnegative integer, ci0,1,2 for all ij, and cj1,2.

aIf HP=4, PJ=5, and PM=2, find LP. _ bIf HP=x+1, PJ=x1, LP=8, and PM=3, find x. _

Solve each of the following equations by finding [ a ]1 and using the result in Exercise 9. a.[ 4 ][ x ]=[ 5 ]in13b.[ 8 ][ x ]=[ 7 ]in11c.[ 7 ][ x ]=[ 11 ]in12d.[ 8 ][ x ]=[ 11 ]in15e.[ 9 ][ x ]=[ 14 ]in20f.[ 8 ][ x ]=[ 15 ]in27g.[ 6 ][ x ]=[ 5 ]in319h.[ 9 ][ x ]=[ 8 ]in242 Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].

For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.