1. If A = [1 rx], B = [ rx P] and C

1. If A = [1 rx], B = [ rx P] and C = [5 b], and if A+yB = C, find the
2 P -1 1 0 1
values of the scalars rx, p, y, b.
2. Locate and correct the two errors in the right-hand side of the equation
o
-1
14
-10
3. The matrices A and B in F. x. are such that A, B, and A + B are all idempotent (i.e.
satisfy X2 = X). Prove that AB = - BA, and deduce, by considering ABA, that AB
=0.
4. The matrices A, B in F. x. are such that both A + B and A - B are idempotent.
Prove that A = A 2 + B2, and deduce that B2 commutes with A. By considering A
= i [x IJ and B = i [ y -IJ' where x and yare numbers to be chosen, show
1 1 -1 -1
that A 2 need not commute with B.
5. The trace of a square matrix means the sum of the entries on its main diagonal.
Prove that, for all A,BEF. x", AB and BA have equal traces. Deduce that if A, BEF.x.
are such that AB - BA is a scalar multiple of I", then AB - BA must be O.
6. In IR. x"' C and D are diagonal matrices, the entries on whose main diagonals are
all positive; and A is such that C2 A = AD2. Prove that CA = AD.
7.* Investigate the circumstances under which the matrix A = [rxik].x. in F.x.
commutes with the n x n matrix unit Eik. Hence or otherwise prove that the only
matrices in F.x. that commute with all matrices in F.x. are the scalar multiples of I •.
S. A square matrix is described as lower triangular if all its entries above the main
diagonal are zero; and if, in addition, all the entries on the main diagonal are zero, the
matrix is described as strictly lower triangular. Show that, in F 3 x 3,
(i) the set of lower triangular matrices is closed under multiplication,
(ii) the cube of every strictly lower triangular matrix is O. (These results generalize to
F.x", with "cube" replaced by "nth power" in (ii).)
9. Let A, B be symmetric matrices in F. x •• Show that:
(i) AB is symmetric if and only if A and B commute;
(ii) AB - BA is skew-symmetric.
10. Let X be a real n x 1 column and S a real n x n skew-symmetric matrix. Show
that the 1 x 1 matrix XTSX is skew-symmetric, and deduce that XTSX = O.
11. In IR. x"' show that the set of orthogonal matrices is (i) closed under
multiplication and (ii)c1osed under the taking of inverses [meaning that if A is in the set,
so also is A-I].
12. In IR. x"' A is an orthogonal matrix and the matrix B is such that AB is skewsymmetric.
Show that (i) BA is also skew-symmetric; (ii) if B is orthogonal, then (AB)2
= (BA)2 = -I.

Kết quả (Việt) 1:

1. If A = [1 rx], B = [ rx P] and C = [5 b], and if A+yB = C, find the2 P -1 1 0 1values of the scalars rx, p, y, b.2. Locate and correct the two errors in the right-hand side of the equationo-114-103. The matrices A and B in F. x. are such that A, B, and A + B are all idempotent (i.e.satisfy X2 = X). Prove that AB = - BA, and deduce, by considering ABA, that AB=0.4. The matrices A, B in F. x. are such that both A + B and A - B are idempotent.Prove that A = A 2 + B2, and deduce that B2 commutes with A. By considering A= i [x IJ and B = i [ y -IJ' where x and yare numbers to be chosen, show1 1 -1 -1that A 2 need not commute with B.5. The trace of a square matrix means the sum of the entries on its main diagonal.Prove that, for all A,BEF. x", AB and BA have equal traces. Deduce that if A, BEF.x.are such that AB - BA is a scalar multiple of I", then AB - BA must be O.6. In IR. x"' C and D are diagonal matrices, the entries on whose main diagonals areall positive; and A is such that C2 A = AD2. Prove that CA = AD.7.* Investigate the circumstances under which the matrix A = [rxik].x. in F.x.commutes with the n x n matrix unit Eik. Hence or otherwise prove that the onlymatrices in F.x. that commute with all matrices in F.x. are the scalar multiples of I •.S. A square matrix is described as lower triangular if all its entries above the maindiagonal are zero; and if, in addition, all the entries on the main diagonal are zero, thema trận được mô tả như là nghiêm thấp hình tam giác. Cho thấy rằng, trong F 3 x 3,(i) các thiết lập của ma trận tam giác thấp hơn đóng cửa dưới phép nhân,(ii) khối của mỗi ma trận tam giác nghiêm ngặt thấp là O. (các kết quả tổng hợp đểF.x", với"khối"thay bằng"quyền lực thứ n"(II).)9. cho A, B là ma trận đối xứng ở F. x •• cho thấy rằng:(i) AB là đối xứng nếu và chỉ nếu A và B đi làm;(ii) AB - BA là skew-symmetric.10. Hãy để X là thực n x 1 cột và S n thực x n ma trận đối xứng skew. Hiển thịlà ma trận 1 x 1 XTSX skew-symmetric, và suy luận rằng XTSX = O.11. ở IR. x"' Hiển thị rằng các thiết lập của ma trận trực giao là (i) đóng cửa dướiphép nhân và (ii) c1osed theo chụp ngược [có nghĩa rằng nếu A là trong các thiết lập,Vì vậy, cũng là một-I].12. ở IR. x"' A là một ma trận trực giao và ma trận B là như vậy mà AB là skewsymmetric.Hiển thị (i) BA cũng là skew-symmetric; (ii) nếu B vuông góc, sau đó (AB) 2= (BA) 2 = - TÔI.