Section

A matrix is a rectangular collection of any number of values, which are called elements or entries. The size or dimension of a matrix is represented by the notation m*n, where m is the number of rows, and n is the number of columns. The rows are numbered from top to bottom, and the columns are numbered from left to right. Each element can be specified by its row and column number with the notation amn, where a is an arbitrary lowercase letter, and the subscripts have the same meaning as they do in the notation m*n . Entire matrices are commonly represented by a capital letter.

A matrix for which m=n is called a square matrix. Square matrices have a diagonal, which consists of all of the elements that are in the m^th row and the m^th column. In other words, the diagonal entries have the same row and column number. A row matrix has only one row (dimension 1*n); a column matrix has only one column (dimension m*n). It is sometimes useful to think of a single constant as a 1*1 matrix.

Examples	Explanation
	3*3 square matrix; diagonal elements are 2, 13, and 2
	2*1 column matrix A

Matrix Addition and Subtraction

Two or more matrices can be combined algebraically. To add matrices, the elements in corresponding positions must be added. For example, the element in the second row, first column of matrix A must be added to the element in the second row, first column of matrix B. The procedure for subtraction is similar. These operations require that the matrices be of the same dimension, otherwise some of the elements of one matrix will have no corresponding entries in the other matrix/matrices.

Examples	Explanation

Special Matrices

One very important is the identity matrix:

The identity matrix I_n (of dimension m*n) is always square, but beyond this feature it can have a size necessary to solve a given problem. Its elements consist of 1’s all along its diagonal, while all of the other elements are zero.

A zero matrix is a matrix of any size in which all of the elements are 0.

Try these exercises

Instructions

1.    What is the size of N?


2.    What are the diagonal entries of M?


3.    What is the row and column number of the number 3 in B?


4. How many 1’s are in I₁₁ ?


5. Is the sum or product of a matrix of dimension 1*5 and a matrix of dimension 5*1 defined?


6.    Find the sum of M and the 2*2 identity matrix.


7.    Find A+B.


8.    Find B-A.


9.    Compare A+B to B+A. What arithmetic property of matrices does this suggest?


10.    Find P-Q+R

Answers to questions:

1. 


2. 


3. 


4. 11


5. No. The matrices must have the exact same dimensions.


6. 


7. 


8. 


9. From question 7,
   
   
   Therefore
   
   suggesting that matrix addition is commutative.


10. Because matrix addition is the same as arithmetic addition for each group of elements, the order of operations applies.