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8.2

Operations with Matrices. 8.2. Matrix Addition and Scalar Multiplication. With matrix addition, you can add two matrices (of the same order ) by adding their corresponding entries. Example 1 – Addition of Matrices. What has to be true about the orders of 2 matrices to be able

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8.2

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  1. Operations with Matrices 8.2

  2. Matrix Addition and Scalar Multiplication • With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries.

  3. Example 1 – Addition of Matrices What has to be true about the orders of 2 matrices to be able to add or subtract them?

  4. Example 2 – Addition of Matrices cont’d • d. The sum of • and • is undefined because A is of order 3  3 and B is of • order 3  2.

  5. Matrix Addition and Scalar Multiplication • In operations with matrices, numbers are usually referred to as scalars • You can multiply a matrix A by a scalar c by multiplying each entry in A by c.

  6. Example 3 – Scalar Multiplication and Matrix Subtraction • For the following matrices, find (a) 3A, (b) –B, and (c) 3A – B. (a) Find 3A

  7. Example 3 – Scalar Multiplication and Matrix Subtraction • For the following matrices, find (a) 3A, (b) –B, and (c) 3A – B. (b) Find -B

  8. Example 3 – Scalar Multiplication and Matrix Subtraction • For the following matrices, find (a) 3A, (b) –B, and (c) 3A – B. (c) Find Find 3A-B

  9. Matrix Multiplication • For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. • That is, the middle two numbers must be the same. The outside two numbers give the order of the product, as shown below. • A  B = AB m p n p m n The INNER dimensions of the two matrices have to be the same in order to multiply them!

  10. Example 4 – Finding the Product of Two Matrices • Find the product AB using and • Solution: • To find the entries of the product, multiply each row of A by each column of B

  11. Finish on Whiteboard • H Dub • 8-2 Pg. 598#5-8all, 33, 34, 41-46all

  12. Matrix Addition and Scalar Multiplication • The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers.

  13. Matrix Addition and Scalar Multiplication • if A is an m n matrix and O is the m nzero • matrix consisting entirely of zeros, then A + O = A. • O is the additive identity for the set of all m n matrices. • For example, the following matrices are the additive identities for the sets of all 2 3 and 2 2 matrices. • and 2 2zero matrix 2 3zero matrix

  14. Example 4 – Solution cont’d

  15. Matrix Multiplication • Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. • That is, the middle two indices must be the same. The outside two indices give the order of the product, as shown below. • A  B = AB m p n p m n

  16. Matrix Multiplication • Even if both AB and BA are defined, matrix multiplication is not, in general, commutative. That is, for most matrices, AB BA. This is one way in which the algebra of real numbers and the algebra of matrices differ.

  17. Matrix Multiplication

  18. Matrix Multiplication • If A is an n  n matrix, the identity matrix has the property that AI = A and IA = A. For example, • and AI = A IA = A

  19. H Dub • 8-2 Pg. 598#5-8all, 33, 34, 41-46all

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