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 A1 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 mAB is invertible if and only if I nBA 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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 Spring '07
 Cheng
 Linear Algebra, Algebra, Addition, Matrices, Diagonal matrix

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