Matrices Worksheet March 9, 2011
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Matrices Worksheet Page 1 of 1 Linear Algebra, Spring 2011 Matrices Worksheet March 9, 2011 (Do your answers on a separate sheet of paper.) 1. Let A be a square matrix. Prove that A is invertible if and only if AT is invertible. 2. Let A be a square matrix and let k 2 R. Prove that (kA)T = k(AT ). 3. Prove that (A−1)T = (AT )−1.(Hint: Compute (A−1)T AT .) (We already saw that (Ar)T = (AT )r if r > 0, but in this problem r < 0.) 4. Suppose A is symmetric and invertible. Prove that A−1 is also symmetric. 5. Let A and B be n × n matrices of rank n. Prove that AB has rank n. 6. Let A and B be n × n matrices, and suppose that the product AB is invertible. Prove that both A and B are also invertible. (Be careful not to beg the question!) (In particular, you cannot use Theorem 5.9, since that assumes A and B are invertible!) 7. We say that an n × n matrix is skew-symmetric if AT = −A. (a) Prove that the main diagonal of a skew-symmetric matrix contains nothing but zeros. (b) Let B be an n × n matrix. Prove that B − BT is skew-symmetric. (c) Let B be an n × n matrix. Prove that B + BT is symmetric. (d) Let B be an n×n matrix. Prove that B can be written as the sum of a symmetric matrix and a skew-symmetric matrix. 0 1 0 1 u1 v1 B u C B v C B 2 C m B 2 C n T XC. (a) Let u = B . C 2 R and let v = B . C 2 R . Then v is a row vector. @ . A @ . A um vn Suppose that u and v are both nonzero. Prove that uvT is an m × n matrix with rank one. (b) Let A be an m × n matrix with rank one. Prove that there exist vectors u 2 Rm and v 2 Rn such that A = uvT . (c) Suppose the matrix B has rank r and is in row echelon form. Prove that B can be written as the sum of r matrices, each of rank 1. (Hint: Use problems 7a and 7b.) (d) Suppose the matrix C has rank r (but is not necessarily in row echelon form). Prove that C can be written as the sum of r matrices, each of rank 1..