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111supp1 - AB is symmetric if and only if AB = BA 6 Let A...

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MATH111 Additional Problems 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. 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. 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 . 4. Show that the equation AB - BA = I has no solutions in n × n matrices with real entries. 5. A matrix A is called symmetric if A = A T . a) Prove that for any matrix A , the matrix AA T is symmetric and that if A is a square matrix then A + A T is symmetric. b) Prove that the inverse of an invertible symmetric matrix is also symmetric. c) Let A adn B be symmetric n × n matrices. Prove that the product
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Unformatted text preview: AB is symmetric if and only if AB = BA . 6. Let A be an n × n matrix with integer entries a ij . Prove that A-1 has integer entries if and only if det A = ± 1. 7. Write the matrix ± 1 2 3 4 ² as a product of elementary matrices, using as few as you can. 8. Find a representation of the complex numbers by real 2 × 2 matrices which is compatible with addition and multiplication. Begin by finding a nice solution to the matrix operation A 2 =-I . 9. Let A,B be m × n and n × m matrices. Prove that I m-AB is invertible if and only if I n-BA is invertible. 10. A n × n matrix A is called unipotent if A is upper triangular and the diagonal entries of A are 1. Prove that every unipotent matrix is invertible and its inverse is also a unipotent matrix. 1...
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