8 1 Matrices Systems of Equations To solve

8. 1 Matrices & Systems of Equations To solve a system of equations 1. Write system of equations as an augmented matrix 2. Use row operations to convert to row echelon form 3. Find variables using back-substitution WE’LL DO EXAMPLES IN CLASS! • Augmented Matrix • Row Operations • Row Echelon Form –> Gaussian Elimination • Reduced Row Echelon Form -> Gauss-Jordan Elimination Augmented Matrix Example 3 x x x + + + y y 3 y + + + 2 z 2 z 2 z = = = 31 19 25 3 1 1 3 2 2 2 31 19 25 Augmented Matrix in Row Echelon Form (Different Example) x + 2 y y – + 5 z 3 z z = -19 = 4 1 0 0 2 1 0 -5 -19 3 9 1 4 * Reduced Row Echelon Form has 1’s along the diagonal and 0’s everywhere else

Matrix Row Operations 1. Two rows of a matrix may be interchanged. 2. The elements in any row may be multiplied by a nonzero number. (3 R 2) 3. Multiply a row by a non-zero number and add to another row (Example: 2 R 1 + R 3 -> R 3) 3 18 -12 21 1 2 -3 5 -2 -3 4 -6 Perform row operations: Interchange: R 1 R 2 1 2 -3 5 3 18 -12 21 -2 -3 4 -6 3 R 1 9 54 -36 63 1 2 -3 5 -2 -3 4 -6 2 R 2 + R 3 3 1 0 18 -12 21 2 -3 5 1 -2 4

8. 2 Matrix Algebra A matrix of order m x n has m rows and n columns A = [aij] (Matrix a with elements aij) Order: 2 x 3 a 23 = -1/5 a 12. = 2 Matrix Addition é 2 - 1 3 ù é- 2 1 0 ù = +ê ê ú ú 4 0 5 1 3 1 ë û Scalar Multiplication 3 3 = é 0 0 3 ê 3 -6 ë 5 é 93 ê ë -12 6 - 15 ù ú - 3/ 5û 0 **Note: A matrix where all elements are zero is known as the zero matrix : 0

Equality of Matrices Two matrices A and B are equal (A = B) if they have the same Order (number of rows and columns are equal) and each (i, j) entry Of A is equal to the corresponding (i, j) entry of B. X– 8 Y+7 6 5 X+Y 2 3 x-y = 10 15 = 4 3 V R 2 1 x – 8 = 10 x = 18 Y + 7 = 15 y = 8 V = 6 and R = 5 x+y=4 X–y=1 So, x = 5/2 and y = 3/2

Matrix Multiplication Note: AB is often not equal to BA Try A = 2 3 X B= 3 4 -1 0 5 1

8. 3 The Matrix Inverse For the Real Number System: A x 1 = A and 1 x A = A So, 1 is the multiplicative Identity For Matrices: A x I = A and I x A = A So, I is the multiplicative Identity Matrix. I is a square matrix (2 x 2, 3 x 3, etc. ) with 1’s in the Diagonal and 0’s elsewhere 5 6 7 8 X 1 0 0 1 = 5 7 6 8 = 1 0 0 1 X 5 7 6 8 For the Real Number System (A) (1/A) = 1 and (1/A)(A) = 1 (Identity) So, 1/A is the multiplicative inverse of A and A is the multiplicative inverse of 1/A For Matrices: AA-1 = I and A-1 A = I (If AB = BA = I then A & B are inverses) -1 3 2 -5 5 2 5 3 2 1 = 1 0 0 1 = 3 1 -1 3 2 -5

Finding a Matrix Inverse or Proving None To prove a matrix has no inverse, suppose has an inverse Find the multiplicative inverse of 2 1 5 3 2 1 w x 1 0 = 5 3 y z 0 1 Then, 2 w + y = 1 5 w + 3 y = 0 2 x + z = 0 5 x + 3 z = 1 Solve the system of equations X = -1 w = 3 So, A-1 = 3 -1 Y = -5 z = 2 -5 2 Since the two matricies are equal y ou Must have 0 = 1, but this is False, Thus, No Inverse. Special rule for inverse of a 2 x 2 matrix Let A= a b If ad – bc =0 the no inverse c d Otherwise, A-1 =

Another Method for Finding an Inverse Start With: Perform Row Operations until The left is the Identity Matrix Thus, the inverse is:

Solving Systems of Equations using Inverse To solve the system, solve The matrix equation : AX = B X = A-1 B Write the linear system in matrix form, then find the inverse. We know the inverse of this matrix since we found it in the previous example. A X = B So, the solution set is {(3, 1, 4)}

8. 4 Determinants and Cramer’s Rule The determinant of a 2 x 2 matrix, A is denoted det(A), |A| or |A| = ad - bc Example: The determinant of a square n x n matrix, A, (n ≥ 3). is the sum of the entries in any row of A (or column of A), multiplied by their respective cofactors. The minor Mij of the element aij is the determinant of the (n– 1) × (n– 1) matrix obtained by deleting the ith row and the jth column of A. The cofactor Aij of the entry aij is given by:

Finding Minors and Cofactors a. Find minors M 11 and M 32 b. Find cofactors A 11 and A 32 c. Find |A| To find M 11 delete the first row and first column Then, find the determinant: (-3)(7) – (-6)(1) = -21 - -6 = -21 + 6 = -15 To find M 32 delete third row and second column Then, find the determinant: (-6)(2) – (4)(5) = -12 - 20 = -32 To find cofactors A 11 & A 32 A 11 = (-1)1+1 (-15) = (-1)2 (-15) = 1(15) = 15 A 32 = (-1)3+2 (-32) = (-1)5 (-32) = -1(-32) = -32 11

Example: Find the Determinant Recall: The determinant of a square n x n matrix, is the sum of the entries in any row (or column), multiplied by their respective cofactors. 12

CRAMER’S RULE FOR SOLVING TWO EQUATIONS IN TWO VARIABLES The system of two equations in two variables has a unique solution (x, y) given by provided that D ≠ 0, where 13 © 2010 Pearson Education, Inc. All rights reserved

Using Cramer’s Rule to Solve Systems of Equations Solve the system (2 x 2): Solve the system(3 x 3) Recall: |A| = a 11 A 11+a 21 A 21+a 31 A 31 © 2010 Pearson Education, Inc. All rights reserved 14