1Chapter 1
1-2
mn m (row) n (column) m n (order)(matrix) A
(1-2)
1-1
1-*
m n A = [aij]mn[aij] A i j aij A (i, j)(element, or entry)
1-1
A m n
m = 1 A (row matrix)(row vector) A = [3, 2, 1] n = 1 A (column matrix)(column vector)
Chapter 1
1-*
m = n A n (square matrix of order n)a11, a22, ……, ann (main diagonal)
n 0aij = 0 i j (diagonal matrix)
Chapter 1
A 1 A (identity matrix) In
A aii = c, i = 1,…, n A (scalar matrix)
Chapter 1
1-*
n A aij = 0 i > j A (upper triangular matrix)
Chapter 1
1-*
aij = 0 i < j A (lower triangular matrix)
(triangular matrix)
0 (zero matrix) O Om n
Chapter 1
1-3
A = [aij] B = [bij] m n aij = bij 1 i m 1 j n A B (equal) A = B
1-2
(matrix addition)
A = [aij] B = [bij] m n A B(sum) C = [cij] m n
C A B m n C = A + B
1-3
(matrix multiplication)
A = [aij] m n B = [bij] n p A B (product) C = [bij] m p
1-4
(2) A + (B + C) = (A + B) + C (associative property)
(3) A + O + O + A = A (identity property) O
1-1
property)
(3) A m n AIn = Im A = A
1
1-2
1-*
r A = [aij] rA A r (scalar multiplication)
A = [aij] m n AT = [aji] n m A (transpose of A)
1-5
1-6
(1) r (sA) = (rs)A
(3) r (A + B) = rA + rB
(4) A (rB) = r (AB) = (rA) B
(5) oA = O
(6) rO = O
(1)(AT)T = A
aij = aji
1-*
X = [x1, x2, ……xn ]T n 1 b = [b1, b2, …… bn]T m 1
(augmented matrix)
(1) r s ( Rr Rs )
(2) r c ( cRr )
(3) r c s ( cRr + Rs )
1-8
1-*
A B A B (row equivalent) A ~ B
A ~ A
A ~ B B ~ C A ~ C
1-9
(1) 1(leading entry) ;
(3)
1-10-2
1-*
A m n C A C A (rank) r (A)
1-11
(1) r (A) = r ([A b]) = n AX = b
(2) r (A) = r ([A b]) < n AX = b
(3) r (A) < r ([A b]) AX = b
1-5
1-*
AX = b [A b] [C d] CX = d AX = b CX = d
[A b][C d]
[C d](Gaussian elimination)
[C d]-(Gauss-Jordan reduction)
1-5-1
AX = 0 (1-4)
b 0(1-3)(nonhomogeneous system of linear equations)
x1 = 0, x2 = 0, ……, xn = 0(1-4)(trivial solution)
xi 0(non-trivial solution)
Chapter 1
1-*
A m n m < n AX = 0
1-6
AX = bA n
B BA = In AX = b B
BAX = Bb
InX = Bb
X = Bb
AX = b X = Bb B A
Chapter 1
1-*
A n n BAB = BA = In A (nonsingular matrix)(invertible matrix) B A B A (singular matrix)(noninvertible matrix)
1-12
B C A
BA = AB = In
CA = AC = In
1-7
1-*
A n A ~ In A ~ In A
A A1
AA1 = In (1-5)
[X1,……,Xn] [E1,……,En]
Chapter 1
n
1-8(1-6)(1-8)
Chapter 1
[A In] ~ [In A1] (1-10)
[A In] [In B] B A
Chapter 1
1-*
(1) A A1 (A1) 1 = A
(2) AB AB (AB) 1 = B 1A 1
(3) A AT (AT) 1 = (A1) T
1-9
1-*

AX = b AA1X = A1 b
A1 (AX) = A1 b
( A1 A)X = A1 b
InX = A1 b
X = A1 b
Chapter 1
Chapter 1
Chapter 1
(2) A |A| = a11a21 a12a22
1-13
1-*
A = [aij] n Mij A i j n1 Mij |Mij| aij (minor)
(1)i + j |Mij| aij (cofactor)Aij
1-14
(1-11)
1-15 |A| i n1 i i Aij , j = 1, ……, n aij j j Aij aij i = 1, ……, n
(1-12)
1-15
(3) A |A| = a11a22 ……ann
(4) A () B |B| = |A|
1-10
(5) A () c B |B| = c |A|
(6) A () c () B |B| = |A|
(7) B n |AB| = |A||B|
(8) |AT| = |A|
(9)(3)(7) A1A = In
1-10
(2) A |A|
()
i k
j k
0
1-*
A = [aij] n adj A = [Aij]T A (adjoint matrix of A)
adj AA A
1-16
1-*
A n |A| 0 A
A n AX = 0 |A| = 0
1-12
1-13
|A |0
(d) Ax=b?

Ai A i b = [b1, b2 ,……,bn]T
i