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Additional info for Schaum’s Outline of Linear Algebra, 4th Edition
Example text
B ¼ ½5, C ¼ 5 7 4 Because M is block diagonal, square each block: ! 7 10 2 ; B2 ¼ ½25; A ¼ 15 22 16 C ¼ 40 2 ! 37. Let f ðxÞ and gðxÞ be polynomials and let A be a square matrix. Prove (a) ð f þ gÞðAÞ ¼ f ðAÞ þ gðAÞ, (b) ð f Á gÞðAÞ ¼ f ðAÞgðAÞ, (c) f ðAÞgðAÞ ¼ gðAÞ f ðAÞ. Suppose f ðxÞ ¼ Pr i¼1 ai xi and gðxÞ ¼ Ps j¼1 bj xj . (a) We can assume r ¼ s ¼ n by adding powers of x with 0 as their coefficients. Then n P f ðxÞ þ gðxÞ ¼ ðai þ bi Þxi i¼1 n n n P P P Hence, ð f þ gÞðAÞ ¼ ðai þ bi ÞAi ¼ ai Ai þ bi Ai ¼ f ðAÞ þ gðAÞ i¼1 i¼1 i¼1 P (b) We have f ðxÞgðxÞ ¼ ai bj xiþj .
55. Find the inverses of A ¼ 1 2 5 and B ¼ 0 1 À1 5. 56. Suppose A is invertible. Show that if AB ¼ AC, then B ¼ C. Give an example of a nonzero matrix A such that AB ¼ AC but B 6¼ C. 57. Find 2 Â 2 invertible matrices A and B such that A þ B 6¼ 0 and A þ B is not invertible. 58. Show (a) A is invertible if and only if AT is invertible. (b) The operations of inversion and transpose commute; that is, ðAT ÞÀ1 ¼ ðAÀ1 ÞT . (c) If A has a zero row or zero column, then A is not invertible. 59. Let A ¼ diagð1; 2; À3Þ and B ¼ diagð2; À5; 0Þ.
Thus, BH ¼ BÀ1 , which means B is unitary. (c) To show C is normal, we evaluate CC H and C H C: ! ! 14 4 À 4i 2 À 3i Ài 2 þ 3i 1 H ¼ CC ¼ 4 þ 4i 6 1 1 À 2i i 1 þ 2i ! 14 4 À 4i and similarly C H C ¼ . Because CC H ¼ C H C, the complex matrix C is normal. 4 þ 4i 6 We note that when a matrix A is real, Hermitian is the same as symmetric, and unitary is the same as orthogonal. 12 Block Matrices Using a system of horizontal and vertical (dashed) lines, we can partition a matrix A into submatrices called blocks (or cells) of A.