Example 3: Scalar Multiplication and Matrix Subtraction It 131 Mathematics For Science Semester 1 2017 IT 131: Mathematics for Science Lecture Notes 3 Source: Larson, Edwards, Falvo

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It 131 Mathematics For Science Semester 1 2017

IT 131: Mathematics for Science

Lecture Notes 3 Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition.

Matrices

2.1 Operations with Matrices

This section and the next introduce some fundamentals of matrix theory.

It is standard mathematical convention to represent matrices in any one of the following three ways.

1. A matrix can be denoted by an uppercase letter such as

A, B, C, . . . .

2. A matrix can be denoted by a representative element enclosed in brackets, such as

aij , bij , cij , . . . .

A matrix can be denoted by a rectangular array of numbers

As mentioned in the previous Lecture, the matrices are primarily real matrices. That is, their entries contain

real numbers.

Two matrices are said to be equal if their corresponding entries are equal.

Definition of Equality of Matrices

Example 1: Equality of Matrices

Consider the four matrices

Matrices A and B are not equal because they are of different sizes. Similarly, B and C are not equal. Matrices

A and D are equal if and only if x= 3.

A matrix that has only one column is called a column matrix or column vector. Similarly, a matrix that has

only one row is called a row matrix or row vector. Boldface lowercase letters are often used to designate column

matrices and row matrices. For instance, matrix A in Example 1 can be partitioned into two column matrices

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MATRIX ADDITION

You can add two matrices (of the same size) by adding their corresponding entries.

Definition of Matrix Addition

Example 2: Addition of Matrices

SCALAR MULTIPLICATION

When working with matrices, real numbers are referred to as scalars. You can multiply a matrix A by a scalar

c by multiplying each entry in A by c.

Definition of Scalar Multiplication

You can use - A to represent the scalar product ( -1)A. If A and B are of the same size,

A - B represents the sum of A and ( - 1)B. That is

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Example 3: Scalar Multiplication and Matrix Subtraction

Solution

MATRIX MULTIPLICATION

The third basic matrix operation is matrix multiplication.

Definition of Matrix Multiplication

This definition means that the entry in the i th row and the j th column of the product AB is obtained by

multiplying the entries in the i th row of A by the corresponding entries in the j th column of B and then

adding the results. The next example illustrates this process.

Example 4: Finding the Product of Two

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Solution

Be sure you understand that for the product of two matrices to be defined, the number of columns of the first

matrix must equal the number of rows of the second matrix. That is,

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So, the product BA is not defined for matrices such as A and B in Example 4

The general pattern for matrix multiplication is as follows. To obtain the element in the i th row and the j th

column of the product AB, use the i th row of A and the j th column of B.

Discovery

Example 5: Matrix Multiplication

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R E M A R K : Note the difference between the two products in parts (d) and (e) of Example 5. In

general, matrix multiplication is not commutative. It is usually not true that the product AB is equal to the

product BA.

SYSTEM OF LINEAR EQUATION

One practical application of matrix multiplication is representing a system of linear equations. Note

how the system

can be written as the matrix equation Ax b, where A is the coefficient matrix of the system, and x and b are

column matrices. You can write the system as

Example 6: Solving a System of Linear Equations

Solution

Using Gauss-Jordan elimination on the augmented matrix of this system, you obtain

So, the system has an infinite number of solutions. Here a convenient choice of a parameter is x3= 7t,

and you can write the solution set as

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In matrix terminology, you have found that the matrix equation

has an infinite number of solutions represented by

That is, any scalar multiple of the column matrix on the right is a solution.

PARTITIONED MATRICES

The system Ax = b can be represented in a more convenient way by partitioning the matrices A and x in the

following manner. If

are the coefficient matrix, the column matrix of unknowns, and the right-hand side, respec- tively, of the m n

linear system Ax = b, then you can write

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In other words,

where a1, a2, . . . , an are the columns of the matrix A. The expression

is called a linear combination of the column matrices a1, a2, . . . , an with coefficients

x1, x2 , . . . , xn.

In general, the matrix product Ax is a linear combination of the column vectors a1, a2,. . . , an that form the

coefficient matrix A. Furthermore, the system Ax = b is consistent if and only if b can be expressed as such a

linear combination, where the coefficients of the linear combination are a solution of the system.

Example 7: Solving a System of Linear Equations

can be rewritten as a matrix equation Ax=b, as follows.

Using Gaussian elimination, you can show that this system has an infinite number of solutions, one of

which is x1 = 1, x2 = 1, x3 = 1.

That is, b can be expressed as a linear combination of the columns of A. This representation of one column

vector in terms of others is a fundamental theme of linear algebra.

Just as you partitioned A into columns and x into rows, it is often useful to consider an m x n matrix partitioned

into smaller matrices. For example, the matrix on the left below can be partitioned as shown below at the right.

The matrix could also be partitioned into column matrices

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or row matrices

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Example 3: Scalar Multiplication and Matrix Subtraction It 131 Mathematics For Science Semester 1 2017 IT 131: Mathematics for Science Lecture Notes 3 Source: Larson, Edwards, Falvo