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111supp1soln - MATH111 Additional Problems Solution The...

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MATH111 Additional Problems: Solution The problems are quite difficult if your goal is just a pass for the midterm. 1. A n × n matrix A is called upper triangular if a ij = 0 whenever i > j . Prove that the product of two upper triangular matrices is upper triangular. Soln: Let A = ( a ij ), B ( b ij ) be two upper triangular matrices. Then when i > j , ( AB ) ij = X k a ik b kj = X k j a ik b kj + X k>j a ik b kj . Since i > j , for k j , a ik = 0. So the first term equals 0. For k > j , b kj = 0. So the second term equals 0. So ( AB ) ij = 0 whenever i > j , which means AB is upper triangular. 2. A square matrix A is called nilpotent if A k = 0 for some k > 0. Prove that if A is nilpotent, then I + A is invertible. Soln: First we let k be the smallest integer such that A k = 0. We want to find D such that D ( I + A ) = ( I + A ) D = I . By guessing the form of D , we treat A as a real number and we do long division. Then we guess D = I - A + A 2 - A 3 + · · · = I - A + A 2 - A 3 + · · · + ( - 1) k - 1 A k - 1 since A k = 0. Then we try ( I + A ) D = ( I + A )( I - A + · · · + ( - 1) k - i A k - 1 = I and D ( I + A ) = ( I + A )( I - A + · · · + ( - 1) k - i A k - 1 = I . 3. The trace of a square matrix is the sum of its diagonal entries: tr A = a 11 + a 22 + · · · + a nn . a) Show that tr( A + B ) = tr A + tr B , and that tr AB = tr BA . b) Show that if B is invertible, then tr A = tr BAB - 1 .
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