1. 259 Lecture 14 Elementary Matrix Theory

2. 2 Matrix Definition  A matrix is a rectangular array of elements (usually numbers) written in rows and columns.  Example 1: Some matrices:

3. 3 Matrix Definition  Example 1 (cont.): Matrix A is a 3 x 2 matrix of integers. A has 3 rows and 2 columns. Matrix B is a 2 x 2 matrix of rational numbers. Matrix C is a 1 x 4 matrix of real numbers. We also call C a row vector. A matrix consisting of a single column is often called a column vector.

4. 4 Matrix Definition  Notation:

5. 5 Arithmetic with Matrices  Matrices of the same size (i.e. same number of rows and same number of columns), with elements from the same set, can be added or subtracted!  The way to do this is to add or subtract corresponding entries!

6. 6 Arithmetic with Matrices

7. 7  Example 2: For matrices A and B given below, find A+B and A-B.

8. 8 Arithmetic with Matrices  Example 2 (cont): Solution:  Note that A+B and A-B are the same size as A and B, namely 2 x 3.

9. 9 Arithmetic with Matrices  Matrices can also be multiplied. For AB to make sense, the number of columns in A must equal the number of rows in B.

10. 10 Arithmetic with Matrices  Example 3: For matrices A and B given below, find AB and BA.

11. 11 Arithmetic with Matrices  Example 3 (cont.):  A x B is a 3 x 2 matrix. To get the row i, column j entry of this matrix, multiply corresponding entries of row i of A with column j of B and add.  Since B has 2 columns and A has 3 rows, we cannot find the product BA (# columns of 1st matrix must equal # rows of 2cd matrix).

12. 12 Arithmetic with Matrices  Another useful operation with matrices is scalar multiplication, i.e. multiplying a matrix by a number.  For scalar k and matrix A, kA=Ak is the matrix formed by multiplying every entry of A by k.

13. 13 Arithmetic with Matrices  Example 4:

14. 14 Identities and Inverses  Recall that for any real number a, a+0 = 0+a = a and (a)(1) = (1)(a) = a.  We call 0 the additive identity and 1 the multiplicative identity for the set of real numbers.  For any real number a, there exists a real number –a, such that a+(-a) = -a+a = 0.  Also, for any non-zero real number a, there exists a real number a -1 = 1/a, such that (a -1 )(a) = (a)(a -1 ) = 1.  We all –a and a -1 the additive inverse and multiplicative inverse of a, respectively.

15. 15 Identities and Inverses  For matrices, we also have an additive identity and multiplicative identity!

16. 16 Identities and Inverses A+0 = 0+A = A and AI = IA = A holds. (HW-check!)

17. 17 Identities and Inverses  Clearly, A+(-A) = -A + A = 0 follows! Note also that B-A = B+(-A) holds for any m x n matrices A and B.

18. 18 Identities and Inverses  Example 5:

19. 19 Identities and Inverses  Example 5 (cont):

20. 20 Identities and Inverses  Example 5 (cont.)

21. 21 Identities and Inverses  Example 5 (cont.)

22. 22 Identities and Inverses  Example 5 (cont):

23. 23 Identities and Inverses  For multiplicative inverses, more work is needed.  For example, here is one way to find the matrix A -1, given matrix A, in the 2 x 2 case!

24. 24 Identities and Inverses

25. 25 Identities and Inverses  From the first matrix equation, we see that e, f, g, and h must satisfy the system of equations:  ae + bg = 1 af + bh = 0 ce + dg = 0 cf + dh = 1.  It follows that if e, f, g, and h satisfy this system, then the second matrix equation above also holds!  Solving the system of equations, we find that ad-bc  0 must hold and e = d/(ad-bc), f = -b/(ad-bc), g = -c/(ad-bc), h = a/(ad-bc).  Thus, we have the following result for 2 x 2 matrices:

26. 26 Identities and Inverses  In this case, we say A is invertible.  If ad-bc = 0, A -1 does not exist and we say A is not invertible.  We call the quantity ad-bc the determinant of matrix A.

27. 27 Identities and Inverses  Example 6: For matrices A and B below, find A -1 and B -1, if possible.

28. 28 Identities and Inverses  Example 6 (cont.)  Solution: For matrix A, ad-bc = (1)(4)-(2)(3)= 4-6 = -2 0, so A is invertible. For matrix B, ad-bc = (3)(2)- (1)(6) = 6-6 = 0, so B is not invertible.  HW-Check that AA -1 = A -1 A = I!!  Note: For any n x n matrix, A -1 exists, provided the determinant of A is non-zero.

29. 29 Linear Systems of Equations  One use of matrices is to solve systems of linear equations.  Example 7: Solve the system x + 2y = 1 3x + 4y = -1  Solution: This system can be written in matrix form AX=b with:

30. 30 Linear Systems of Equations  Example 7 (cont.)  Since we know from Example 6 that A -1 exists, we can multiply both sides of AX = b by A -1 on the left to get: A -1 AX = A -1 b => X = A -1 b.  Thus, we get in this case:

31. 31 Linear Systems of Equations  Example 7 (cont.):

32. 32 References  Elementary Linear Algebra (4 th ed) by Howard Anton.  Cryptological Mathematics by Robert Edward Lewand (section on matrices).