Key Words
3.1
Matrices
Copyright © Cengage Learning. All rights reserved.
Matrices
Matrices are classified in terms of the numbers of rows and columns they have. Matrix M has three rows and four columns, so we say this is a 3 4 (read “three by four”) matrix. 2
Matrices
The matrix
has m rows and n columns, so it is an m n matrix. When we designate A as an m n matrix, we are indicating the size of the matrix.
3
Matrices
Two matrices are said to have the same order (be the same size) if they have the same number of rows and the same number of columns.
For example,
and
do not have the same order.
C is a 2 3 matrix and D is a 3 2 matrix.
4
Matrices
The numbers in a matrix are called its entries or elements.
Note that the subscripts on an entry in matrix A above correspond respectively to the row and column in which the entry is located.
Thus a23 represents the entry in the second row and the third column, and we refer to it as the “two-three entry.” In matrix B below, the entry denoted by b23 is 1.
Some matrices take special names because of their size. If the number of rows equals the number of columns, we say the matrix is a square matrix.
5
Matrices
Matrix B below is a 3 3 square matrix.
A matrix with one row, such as [9 5] or [3 2 1 6], is called a row matrix, and a matrix with one column, such as
is called a column matrix.
6
Matrices
Row and column matrices are also called vectors.
Any matrix in which every entry is zero is called a zero matrix; examples include
and
We define two matrices to be equal if they are of the same order and if each entry in one equals the corresponding entry in the other.
7
Matrices
When the columns and rows of matrix A are interchanged to create a matrix B, and vice versa, we say that A and B are transposes of each other and write AT = B and BT = A.
8
Example 2 – Matrices
(a) Which element of
is represented by a32?
(b) Is A a
Matrices
Copyright © Cengage Learning. All rights reserved.
Matrices
Matrices are classified in terms of the numbers of rows and columns they have. Matrix M has three rows and four columns, so we say this is a 3 4 (read “three by four”) matrix. 2
Matrices
The matrix
has m rows and n columns, so it is an m n matrix. When we designate A as an m n matrix, we are indicating the size of the matrix.
3
Matrices
Two matrices are said to have the same order (be the same size) if they have the same number of rows and the same number of columns.
For example,
and
do not have the same order.
C is a 2 3 matrix and D is a 3 2 matrix.
4
Matrices
The numbers in a matrix are called its entries or elements.
Note that the subscripts on an entry in matrix A above correspond respectively to the row and column in which the entry is located.
Thus a23 represents the entry in the second row and the third column, and we refer to it as the “two-three entry.” In matrix B below, the entry denoted by b23 is 1.
Some matrices take special names because of their size. If the number of rows equals the number of columns, we say the matrix is a square matrix.
5
Matrices
Matrix B below is a 3 3 square matrix.
A matrix with one row, such as [9 5] or [3 2 1 6], is called a row matrix, and a matrix with one column, such as
is called a column matrix.
6
Matrices
Row and column matrices are also called vectors.
Any matrix in which every entry is zero is called a zero matrix; examples include
and
We define two matrices to be equal if they are of the same order and if each entry in one equals the corresponding entry in the other.
7
Matrices
When the columns and rows of matrix A are interchanged to create a matrix B, and vice versa, we say that A and B are transposes of each other and write AT = B and BT = A.
8
Example 2 – Matrices
(a) Which element of
is represented by a32?
(b) Is A a