By E. S. Venttsel'

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6. R/ is symmetric. R/ is symmetric, then A is symmetric? 7. R/ ! R/ and all c 2 R. 8. R/ be a matrix such that A t A D On . Prove that A D On . 9. R/ such that A2 D On . 10. R/ be matrices such that t A1 A1 C : : : C t Ak Ak D On : Prove that A1 D : : : D Ak D On . 11. R/ be a skew-symmetric matrix. Prove that there exists a nonzero vector X 2 R3 such that AX D 0. R/? 12. R/ such that A t A D t A A: Chapter 2 Square Matrices of Order 2 Abstract The main topic of this chapter is a detailed study of 2 2 matrices and their applications, for instance to linear recursive sequences and Pell’s equations.

C/ is a diagonal matrix whose diagonal entries are pairwise distinct. C/ be a matrix such that AB D BA. Prove that B is diagonal. 16. R/ is called a permutation matrix if each row and column of A has an entry equal to 1 and all other entries equal to 0. Prove that the product of two permutation matrices is a permutation matrix. 17. Consider a permutation of 1; 2; : : : ; n, that is a bijective map W f1; 2; : : : ; ng ! j / and 0 otherwise. a) Prove that any permutation matrix is of the form P for a unique permutation .

B) The result does no longer hold. C/ such that t A A D O2 , that is A2 D O2 . Since A is symmetric, we can write Ä AD ab bd for some complex numbers a; b; d . 6 The Transpose of a Matrix 51 It suffices to ensure that a C d D 0 and a2 C b 2 D 0. For instance, one can take a D i , b D 1, d D i . 62. 52 in order to solve part a). Indeed, for any X 2 Rn we have t X. ai1 x1 C ai2 x2 C : : : C ai n xn /2 D 0 iD1 for any choice of real numbers x1 ; : : : ; xn . Since the sum of squares of real numbers equals 0 if and only if all these numbers are equal to 0, we deduce that ai1 x1 C : : : C ai n D 0 for all i 2 f1; 2; : : : ; ng and all real numbers x1 ; : : : ; xn .