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Tổng hợp kiến thức Toán Cao Cấp (AAAclass)
University: Trường Đại học Ngoại thương
Course: Toán cao cấp
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1
TOÁN CAO CẤP
CHƯƠNG I. MÔ HÌNH TUYẾN TÍNH và ĐẠI SỐ MA TRẬN
I.1. MA TRẬN
1. KHÁI NIỆM
a) Ma trận cấp
m n
: là một bảng gồm
mn
số
ij
a
được sắp xếp thành
m
dòng
và
n
cột dưới dạng
11 12 1
21 22 2
1 2
…
…
… … … …
…
n
n
m m mn
a a a
a a a
A
a a a
ij
a
là phần tử ở dòng
i
và cột
j
của ma trận
A
;
i
là chỉ số dòng,
j
là chỉ số cột
của phần tử
ij
a
đó.
Ví dụ 1.
1 2 3
456
A
là ma trận cấp
2 3
, trong ma trận này ta có phần tử
13 21
3, 4
a a
.
Người ta thường viết tắt ma trận ở dạng ij
m n
A a
.
b) Ma trận vuông: là ma trận có số dòng
m
bằng số cột
n
, khi đó thay vì nói ma
trận cấp
n n
ta chỉ nói đó là ma trận vuông cấp
n
.
Ví dụ 2.
1 3
5 7
B
là ma trận vuông cấp hai.
Trong ma trận vuông cấp
n
, người ta gọi các phần tử 11 22
, ,…,
nn
a a a
là các phần
tử thuộc đường chéo chính của ma trận.
c) Ma trận đơn vị: là ma trận vuông có tất cả các phần tử thuộc đường chéo chính
đều bằng 1, các phần tử còn lại đều bằng 0,kí hiệu là
n
I
.
Ví dụ 3. 2 3
1 0 0
1 0
, 0 1 0
0 1
0 0 1
I I
là các ma trận đơn vị cấp 2, cấp 3
d) Ma trận tam giác: là ma trận vuông có tất cả các phần tử nằm phía dưới, hoặc phía
trên đường chéo chính đều bằng 0.
Ví dụ 4.
1 0 0 0
1 2 3
2 3 0 0
0 4 5 ,
4 5 6 0
0 0 6
7 8 9 10
C D
là các ma trận tam giác.
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