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Homework 2 Solution

Homework 2 Solution - MATH 74 HOMEWORK 2 SOLUTIONS 1(a Five...

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MATH 74 HOMEWORK 2 SOLUTIONS 1(a). Five ways. (( a + b ) + c ) + d ( a + b ) + ( c + d ) ( a + ( b + c )) + d a + (( b + c ) + d ) a + ( b + ( c + d )) 1(b). Fourteen ways. a + ( b + ( c + ( d + e ))) a + ( b + (( c + d ) + e )) a + (( b + ( c + d )) + e ) ( a + ( b + ( c + d ))) + e a + (( b + c ) + ( d + e )) a + ((( b + c ) + d ) + e ) ( a + (( b + c ) + d )) + e ( a + ( b + c )) + ( d + e ) (( a + ( b + c )) + d ) + e ( a + b ) + ( c + ( d + e )) ( a + b ) + (( c + d ) + e ) (( a + b ) + ( c + d )) + e (( a + b ) + c ) + ( d + e ) ((( a + b ) + c ) + d ) + e There are various ways of making this listing process more systematic; feel free to ask me about it in office hours. The number of ways of grouping seven numbers is 42. (For more on the sequence 2, 5, 14, 42, . . . , Google “Catalan numbers.”) 2. For any x, y, z we have ( x * y ) * z = ( x · y + x + y ) * z by definition = ( x · y + x + y ) · z + ( x · y + x + y ) + z by definition = x · y · z + x · z + y · z + x · y + x + y + z by D = x · y · z + x · y + x · z + y · z + x + y + z by A2 a few times = x · ( y · z + y + z ) + y · z + x + y + z by D = x · ( y · z + y + z ) + x + y · z + y + z by A2 = x · ( y * z ) + x + ( y * z ) by definition = x * ( y * z ) by definition 3. For any x and y we have x * y = x · y + x + y by definition = y · x + y + x using M2 and A2 = y * x by definition
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2 MATH 74 HOMEWORK 2 SOLUTIONS 4. We have trace( XY ) = trace a b c d p q r s by definition of X, Y = trace ap + br aq + bs cp + dr cq + ds by matrix mult. = ap + br + cq + ds by definition of trace = pa + qc + rb + sd by A2 and M2 = trace pa + qc pb + qd ra + sc rb + sd by definition of trace = trace p q
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