Page 207 - Advanced engineering mathematics

MATRICES ELEMENTARY ROW OPERATIONS
REDUCED ROW ECHELON FORM ROW AND
CHAPTER 7 COLUMN SPACES HOMOGENEOUS SYSTEMS

NONHOMOGENEOUS SYSTEMS MATRIX
Matrices and

Linear Systems

7.1 Matrices

An n by m (or n × m) matrix is a rectangular array of objects arranged in n rows and m
columns.

We will denote matrices in boldface. For example,

2 1 π
A = √
1 2 −5
is a 2 × 3 matrix (two rows, three columns) and
t
e 1 −1 cos(t)
B =
0 4t −7 1 − t
is a 2 × 4 matrix.
The object located in the row i and column j place of a matrix is called its i, j element.
Often we write A =[a ij ], meaning that the i, j element of A is a ij . In the above matrices A and
√
B, a 11 = 2, a 22 = 2, a 23 =−5, b 14 = cos(t) and b 21 = 0.
If the elements of an n × m matrix are real numbers, then each row can be thought of as a
vector in R and each column as a vector in R . In the first example, A has two rows that are
n
m
vectors in R and columns forming three vectors in R . This vector point of view is often useful
3
2
in dealing with matrices.
Two matrices A =[a ij ] and B =[b ij ] are equal if they have the same number of rows, the
same number of columns, and for each i and j, a ij = b ij . Equal matrices have the same
dimensions, and objects located in the same positions in the matrices must be equal.

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October 14, 2010 14:23 THM/NEIL Page-187 27410_07_ch07_p187-246