259 Lecture 14 Elementary Matrix Theory Matrix Definition

259 Lecture 14 Elementary Matrix Theory

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

Matrix Definition o Example 1 (cont. ): n n n 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. 3

Matrix Definition o Notation: 4

Arithmetic with 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! o The way to do this is to add or subtract corresponding entries! 5

Arithmetic with Matrices 6

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

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

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

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

Arithmetic with Matrices o o 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. o Since B has 2 columns and A has 3 rows, we cannot find the product BA (# columns of 1 st matrix must equal # rows of 2 cd matrix). 11

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

Arithmetic with Matrices o Example 4: 13

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

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

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

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

Identities and Inverses o Example 5: 18

Identities and Inverses o Example 5 (cont): 19

Identities and Inverses o Example 5 (cont. ) 20

Identities and Inverses o Example 5 (cont. ) 21

Identities and Inverses o Example 5 (cont): 22

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

Identities and Inverses 24

Identities and Inverses o o o 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: 25

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

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

Identities and Inverses o Example 6 (cont. ) o 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. o HW-Check that AA-1 = A-1 A = I!! o Note: For any n x n matrix, A-1 exists, provided the determinant of A is non-zero. 28

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

Linear Systems of Equations o Example 7 (cont. ) o 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. o Thus, we get in this case: 30

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

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