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By Seymour Lipschutz

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Linear algebra is a starting place path for college students getting into arithmetic, engineering, and machine technology, and the fourth version comprises extra difficulties hooked up at once with purposes to those majors. it's also up-to-date all through to incorporate new crucial appendices in algebraic platforms, polynomials, and matrix applications.

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B ¼ ½5Š, C ¼ 5 7 4 Because M is block diagonal, square each block: ! 7 10 2 ; B2 ¼ ½25Š; 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.

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