1. Meeting 18 Matrix Operations

2. 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 a ij and is called the (i, j)-entry of A. Matrix is a rectangular array of numbers

3. The diagonal entries in an m x n matrix A =[a ij ]  are a 11, a 22, a 33,..., and they form the main diagonal of A. A diagonal matrix is a square n x n matrix whose nondiagonal entries are zero. An example is the n x n identity matrix, I n. An m x n matrix whose entries are all zero is a zero matrix and is written as 0.

4. Sums and Scalar Multiples Two 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 sums of the corresponding columns in A and B. The sum A + B is defined only when A and B are the same size.

5. Example Let

6. 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

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

8. Matrix Multiplication

9. Example Find the product AB where Solution: The product AB has size and will take the form

10. To find c 11 (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 c 12, multiply corresponding entries in the first row of A and the second column of B to obtain

11. Continuing this pattern produces the following results. The product is

12. Examples

14. Properties of Matrix Multiplication

15. Powers of a Matrix If A is an n x n matrix and if k is a positive integer, then A k denotes the product of k copies of A:

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

17. Exercises