SECTION 2.3Arithmetic Operations on Matrices

2.3 Arithmetic Operations on Matrices

1. Definition of Matrix2. Column, Row and Square Matrix3. Addition and Subtraction of Matrices4. Multiplying Row Matrix to Column Matrix5. Matrix Multiplication6. Identity Matrix7. Matrix Equation

Definition of Matrix

A matrix is any rectangular array of numbers and may be of any size.

The size of a matrix is nxk where n is the number of rows and k is the number of columns.

The entry aij refers to the number in the ith row and jth column of the matrix.

Two matrices are equal provided that they have the same size and that all their corresponding entries are equal.

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Example Definition of Matrix

is a 2x3 matrix. The entry a1,2 = -1. The entry a2,3 = 7.

4 1 5

3 0 7

3 9 3 9

7 0 7 0

2 2 2 2

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3 93 7 2

7 09 0 2

2 2

Column, Row and Square Matrix

A row matrix or row vector only has one row.

A column matrix or column vector only has one column.

A square matrix has the same number of rows as columns.

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Example Column, Row & Square Matrix

is a 2x2 matrix and a square matrix.4 1

3 7

6

2 8 7 1 is a 1x4 matrix and a row matrix.

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123.4

is a 3x1 matrix and a column matrix.

Addition and Subtraction of Matrices

The sum A + B of two matrices A and B is defined only if A and B are two matrices of the same size. In this case A + B is the matrix formed by adding the corresponding entries of A and B.

Two matrices of the same size are subtracted by subtracting corresponding entries.

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Example Addition & Subtraction

2 1 3 1 4 7

4 0 5 8 3 2

1 3 10

12 3 3

1 82 1 3

4 34 0 5

7 2

2 1 3 1 4 7

4 0 5 8 3 2

8

3 5 4

4 3 7

is not defined.

Multiplying Row Matrix to Column Matrix

If A is a row matrix and B is a column matrix, then we can form the product AB provided that the two matrices have the same length. The product AB is a 1x1 matrix obtained by multiplying corresponding entries of A and B and then forming the sum.

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21 2 1 1 2 2n n n

n

b

ba a a a b a b a b

b

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Example Multiplying Row to Column

3

2 1 3 2

5

2 3 1 2 3 5 7

3

4 0 2 1 2

5

10

is not defined.

Matrix Multiplication

If A is an mxn matrix and B is an nxq matrix, then we can form the product AB. The product AB is an mxq matrix whose entries are obtained by multiplying the rows of A by the columns of B. The entry in the ith row and jth column of the product AB is formed by multiplying the ith row of A and jth column of B.

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Example Matrix Multiplication

3 2 02 1 3

2 1 23 0 2

5 3 1

3 2 02 1 3

2 1 23 0 2

5 3 1

12

7 12 -5

-19 0 2

is not defined.

Identity Matrix

The identity matrix In of size n is the nxn square matrix with all zeros except for ones down the upper-left-to-lower-right diagonal.

Here are the identity matrix of sizes 2 and 3:

2 3

1 0 01 0

0 1 0 .0 1

0 0 1

I I

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Example Identity Matrix

1 0 2 1 3

0 1 3 0 2

2 1 3

3 0 2

1 0 02 1 3

0 1 03 0 2

0 0 1

2 1 3

3 0 2

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For all nxn matrices A, In A = A In = A.

Matrix Equation

The matrix form of a system of linear equations is

AX = B where A is the coefficient matrix whose

rows correspond to the coefficients of the variables in the equations. X is the column matrix corresponding to the variables in the system. B is the column matrix corresponding to the constants on the right-hand side of the equations.

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Example Matrix Equation

4 7 9

3 2 5.

x y

x y

4 7 9

3 2 5

x

y

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Write the following system as a matrix equation

Equation 1

Equation 2

x y constants

Summary Section 2.3 - Part 1

A matrix of size mxn has m rows and n columns.

Matrices of the same size can be added (or subtracted) by adding (or subtracting) corresponding elements.

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Summary Section 2.3 - Part 2

The product of an mxn and an nxr matrix is the mxr matrix whose ijth element is obtained by multiplying the ith row of the first matrix by the jth column of the second matrix. (The product of each row and column is calculated as the sum of the products of successive entries.)

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