Meeting 18

Matrix Operations

Matrix

If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and j th column of A is denoted by aij and is called the (i, j)-entry of A.

Matrix is a rectangular array of numbers

The diagonal entries in an m x n matrix A =[aij] are � a11, a22, a33, ..., and they form the main diagonal of A.

A diagonal matrix is a square n x n matrix whosenondiagonal entries are zero.

An example is the n x n identity matrix, In.

An m x n matrix whose entries are all zero is a zero matrix and is written as 0.

Sums and Scalar MultiplesTwo matrices are equal if they have the same size (i.e., the same number of rows and the same number of columns) and if their corresponding columns are equal, which amounts to saying that their corresponding entries are equal.

If A and B are m x n matrices, then the sum A + B is the m x n matrix whose columns are the sumsof the corresponding columns in A and B.

The sum A + B is defined only when A and B are the same size.

Example

Let

Scalar Multiple

If r is a scalar and A is a matrix, then the scalar multiple rA is the matrix whose entries are r times the entries in A. As with vectors, -A stands for (-1)A, and A - B is the same as A + (-1)B.

The properties of sums and Scalar Multiplies

Example

If A and B are the matrices in the previous example, then

Matrix Multiplication

Example

Find the product AB where

Solution:

The product AB has size and will take the form

To find c11 (the entry in the first row and first column of the product), multiply corresponding entries in the first row of A and the first column of B. That is,

Similarly, to find c12, multiply corresponding entries in the first row of A and the second column of B to obtain

Continuing this pattern produces the following results.

The product is

Examples

Examples

Properties of Matrix Multiplication

Powers of a MatrixIf A is an n x n matrix and if k is a positive integer, then Ak denotes the product of k copies of A:

The Transpose of a MatrixGiven an m x n matrix A, the transpose of A is the n x m matrix, denoted by AT , whose columns are formed from the corresponding rows of A.

Exercises