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This lesson introduces the matrix - the rectangular array at the heart of matrix algebra. Matrix algebra is used quite a bit in advanced statistics, largely because it provides two benefits.

- Compact notation for describing sets of data and sets of equations.
- Efficient methods for manipulating sets of data and solving sets of equations.

A **matrix** is a rectangular array of numbers arranged in rows and columns. The array of numbers below is an example of a matrix.

The number of rows and columns that a matrix has is called its **dimension** or its **order**. By convention, rows are listed first; and columns, second. Thus, we would say that the dimension (or order) of the above matrix is 3 x 4, meaning that it has 3 rows and 4 columns.

Numbers that appear in the rows and columns of a matrix are called **elements** of the matrix. In the above matrix, the element in the first column of the first row is 21; the element in the second column of the first row is 62; and so on.

Statisticians use symbols to identify matrix elements and matrices.

**Matrix elements.**Consider the matrix below, in which matrix elements are represented entirely by symbols.

- By convention, first subscript refers to the row number; and the second subscript, to the column number. Thus, the first element in the first row is represented by
*A*_{1}_{1}. The second element in the first row is represented by*A*_{1}_{2}. And so on, until we reach the fourth element in the second row, which is represented by*A*_{2}_{4}. **Matrices.**There are several ways to represent a matrix symbolically. The simplest is to use a boldface letter, such as**A**,**B**, or**C**. Thus,**A**might represent a 2 x 4 matrix, as illustrated below.

Another approach for representing matrix

**A**is:**A**= [*A*_{i}_{j}] where i = 1, 2 and j = 1, 2, 3, 4This notation indicates that

**A**is a matrix with 2 rows and 4 columns. The actual elements of the array are not displayed; they are represented by the symbol*A*_{i}_{j}.

Other matrix notation will be introduced as needed. For a description of all the matrix notation used in this tutorial, see the Matrix Notation Appendix.## Matrix Equality

To understand matrix algebra, we need to understand matrix equality. Two matrices are equal if all three of the following conditions are met:

- Each matrix has the same number of rows.
- Each matrix has the same number of columns.
- Consider the three matrices shown below.
- Corresponding elements within each matrix are equal.

If**A**=**B**, we know that x = 222 and y = 333; since corresponding elements of equal matrices are also equal. And we know that matrix**C**is not equal to**A**or**B**, because**C**has more columns than**A**or**B**.## Test Your Understanding

**Problem 1**The notation below describes two matrices - matrix

**A**and matrix**B**.**A**= [*A*_{i}_{j}]where i = 1, 2, 3 and j = 1, 2

Which of the following statements about

**A**and**B**are true?I. Matrix

**A**has 5 elements.

II. The dimension of matrix**B**is 4 x 2.

III. In matrix**B**, element*B*_{2}_{1}is equal to 222.(A) I only

(B) II only

(C) III only

(D) All of the above

(E) None of the above**Solution**The correct answer is (E).

Matrix**A**has 3 rows and 2 columns; that is, 3 rows, each with 2 elements. This adds up to 6 elements, altogether - not 5. The dimension of matrix

**B**is 2 x 4 - not 4 x 2. That is, matrix**B**has 2 rows and 4 columns - not 4 rows and 2 columns.And, finally, element B

_{2}_{1}refers to the first element in the second row of matrix**B**, which is equal to 555 - not 222.

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