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259 Lecture 14

259 Lecture 14

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259 Lecture 14

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  1. 259 Lecture 14 Elementary Matrix Theory

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

  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. Matrix Definition • Notation:

  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. Arithmetic with Matrices

  7. Arithmetic with Matrices • Example 2: For matrices A and B given below, find A+B and A-B.

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

  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. 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. Arithmetic with Matrices • Example 4:

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

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

  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. Identities and Inverses • Example 5:

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

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

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

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

  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. Identities and Inverses

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

  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-1A = I!! • Note: For any n x n matrix, A-1 exists, provided the determinant of A is non-zero.

  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. 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-1AX = A-1b => X = A-1b. • Thus, we get in this case:

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

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