Section 4.1

1. Section 4.1 Using Matrices to Represent Data

2. Matrix Terminology • A matrix is a rectangular array of numbers enclosed in a single set of brackets. • The dimensions of a matrix are the number of horizontal rows and the number of vertical columns it has. • For example, if a matrix has 2 rows and 3 columns, its dimensions are 2 x 3. • Each number in the matrix is an called an entry or element.

3. Using Matrices to Represent Data The table below shows business activity for one month in a home-improvement store. The table shows stock (inventory on June 1), sales (during June), and receipt of new goods (deliveries in June).

4. Examples of Matrices Inventory Matrix Sales Matrix Small Large Picnic tables ⌈ 79 ⌉ Barbeque grills ⌊ 15 12 ⌋ ⌈ 79 ⌉ ͟ S͟ ⌈s₁₁ s₁₂⌉ ⌊ 15 12 ⌋ ͞ ͞ ⌊s₂₁ s₂₂⌋ S is the name of the matrix. Small Large Picnic tables ⌈ 8 10 ⌉ Barbeque grills ⌊ 15 12 ⌋ ⌈ 8 10 ⌉ ͟ M ͟ ⌈m₁₁ m₁₂⌉ ⌊ 15 12 ⌋ ͞ ͞ ⌊m₂₁ m₂₂⌋ M is the name of the matrix.

5. If Two Matrices are Equal • Two matrices are equal if they have the same dimensions and if corresponding entries are equivalent. • Solve ⌈2x+4 51⌉ ͟ ⌈1251⌉ for x & y. ⌊ -2 -3y + 5 -4⌋ ͞ ⌊-2 5y – 3 -4⌋ Because the matrices are equal: 2x + 4 = 12 and -3y + 5 = 5y – 3.

6. Solving for x and y 2x + 4 = 12 Subtract 4 from both sides 2x = 8 Divide both sides by 2 X = 4 -3y + 5 = 5y – 3 Add 3y to both sides 5 = 8y – 3 Add 3 to both sides 8 = 8y Divide both sides by 8 1 = y

7. Addition and Scalar Multiplication • To find the sum (or difference) of matrices A and B with the same dimensions, find the sums (or differences) of corresponding entries in A and B. • Scalar multiplication is multiplication of each entry in a matrix by the same real number.

8. Properties of Matrix Addition • For matrices A, B, and C, each with dimensions of m x n: • Commutative A + B = B + A • Associative (A + B) + C = A + (B + C) • Additive Identity The m x n matrix having 0 as all of its entries is the m x n identity matrix for addition. • Additive Inverse For every m x n matrix A, the matrix whose entries are the opposite of those in A is the additive inverse of A.

9. Section 4.2 Matrix Multiplication

10. Matrix Multiplication • If matrix A has dimensions m x n and matrix B has dimensions n x r, then the product AB has dimensions m x r. • Find the entry in row i and column j of AB by finding the sum of the products of the corresponding entries in row i of A and column j of B.

11. Matrix Multiplication • If matrix A has the dimensions 2 x 3 and matrix B has the dimensions 3 x 2, then: • AB will have the dimensions 2 x 2. • BA will have the dimensions 3 x 3. • For AB, 2 x 3 3 x 2, the 2’s are the outer dimensions and the 3’s are the inner dimensions. • For BA, 3 x 2 2 x 3, the 3’s are the outer dimensions and the 2’s are the inner dimensions.

12. Matrix Multiplication • If the inner dimensions are the same, then multiplication can occur. The outer dimensions give the product dimensions after multiplication occurs. • If the inner dimensions are not the same, then multiplication cannot occur.

13. Matrix Multiplication ⌈ 2 - 3⌉ ⌈5 0⌉ Let R = | 0 5| and W = ⌊4 7⌋ 3x2 ⌊ - 2 0⌋ 2x2 Row 1 of R, Column 1 of W Row 1 of R, Column 2 of W ⌈(2)(5) + (-3)(4) (2)(0) + (-3)(7)⌉ Row 2 of R, Column 1 of W Row 2 of R, Column 2 of W |(0)(5) + (5)(4) (0)(0) + (5)(7)| Row 3 of R, Column 1 of W Row 3 of R, Column 2 of W ⌊(-2)(5) + (0)(4) (-2)(0) + (0)(7)⌋

14. Matrix Multiplication ⌈ - 2 - 21⌉ RW = | 20 35| ⌊ - 10 0⌋ WR - does not exist because the inner dimensions do no not match. W 2 x 2 3 x 2 R

15. Matrix Multiplication • A network is a finite set of connect points. • Each point is called a vertex. • A directed network is a network in which permissible directions of travel between the vertices are indicated. • You can represent a network in an adjacency matrix, which indicates how many one-stage (direct) paths are possible from one vertex to another.

16. Section 4.3 The Inverse of a Matrix

17. Square Matrix • A square matrix is a matrix that has the same number of columns and rows. 2x2, 3x3, 4x4,… • An identity matrix, called I, has 1’s on the main diagonal and 0’s elsewhere. ⌈ 1 0 0⌉ • I₃ₓ₃ = | 0 1 0| ⌊ 0 0 1⌋

18. More About Matrices The Identity Matrix of Mult. The Inverse of a Matrix Let A be a square matrix with n rows and n columns. If there is an n x n matrix B, such that AB = I, then A and B are inverses of one another. The inverse of matrix A is denoted by A⁻ ¹. Note A⁻ ¹ ≠ 1/A • Let A be a square matrix with n rows and n columns. Let Ibe a matrix with the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. Then AI = IA = A

19. Determinant of a 2 x 2 Matrix • Let A = ⌈a b⌉. ⌊c d⌋ The determinant of A, denoted by det(A) or |a b| |c d|, Is defined as det(A) = |a b| = ad – bc. |c d| Matrix A has an inverse if and only if det(A) ≠ 0.

20. Section 4.4 Solving Systems With Matrix Equations

21. Solving Systems With Matrix Equations • A matrix equation – an equation of the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. • A system of linear equations can be used to represent situations and be written as a matrix equation.

22. Solving a Matrix Equation Real Numbers Matrices AX = B A⁻ ¹(AX) = A⁻ ¹(B) (A⁻ ¹)(A)X = A⁻ ¹(B) I X = A⁻ ¹(B) X = A⁻ ¹(B) ax = b (1/a)(ax) = (1/a)(b) (1/a)(a)x = b/a x = b/a A linear equation of the form ax = b, where a, b, and x are real numbers and a≠ 0.

23. Example 5x + 2y – z = - 7 ⌈5 2 - 1⌉ ⌈x⌉ ⌈- 7⌉ x - 2y + 2z = 0 → |1 - 2 2||y| = | 0 | 3y + z = 17 ⌊ 0 3 1⌋ ⌊z⌋ ⌊17⌋ ⌈x⌉ ⌈5 2 - 1⌉⁻ ¹ ⌈- 7⌉ |y| = |1 - 2 2| | 0 | ⌊z⌋ ⌊ 0 3 1⌋ ⌊17⌋ ⌈x⌉ ⌈- 2⌉ |y| = | 4 | ⌊z⌋ ⌊ 5 ⌋ Thus, the solution is x = - 2, y = 4, and z = 5.

24. Section 4.5 Using Matrix Row Operations

25. Using Matrix Row Operations • The row-reduction method of solving a system allows you to determine whether the system is independent, dependent, or inconsistent. • The row-reduction method is performed on an augmented matrix. An augmented matrix consists of the coefficients and constant terms in the system of equations. • Reduced row-echelon form – an augmented matrix is in this form if the coefficient columns form an identity matrix.

26. Elementary Row Operations • The following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form. • Interchange two rows. • Multiply all entries in one row by a nonzero #. • Add a multiple of one row to another.

27. Example System Augmented Matrix m + a + n = 21 ⌈1 1 1 : 21⌉ 2m + a = 23 |2 1 0 : 23| a + 3n = 25 ⌊0 1 3 : 25⌋ coefficients constants -2R₁ + R₂ → R₂ ⌈1 1 1 : 21⌉ |0 -1 -2 : -19| ⌊0 1 3 : 25⌋

28. Example Cont. R₂ + R₁ → R₁ -1 R₂ → R₂ ⌈1 0 - 1: 2 ⌉ ⌈1 0 - 1: 2 ⌉ |0 -1 -2 : -19| |0 1 2 : 19| ⌊0 1 3 : 25⌋⌊0 1 3 : 25⌋ -1 R₂ + R₃ → R₃ R₃ + R₁ → R₁ ⌈1 0 -1 : 2 ⌉ ⌈1 0 0 : 8 ⌉ |0 1 2 : 19| |0 1 2 : 19| ⌊0 0 1 : 6 ⌋ ⌊0 0 1 : 6 ⌋

29. Example Cont. -2R₃ + R₂ → R₂ ⌈1 0 0 : 8 ⌉ |0 1 0 : 7 | ⌊0 0 1 : 6 ⌋ The matrix is now in reduced row-echelon form. m = 8, a = 7, n = 6

30. Dependent, Independent, or Inconsistent • Dependent System – a system of equations that has infinitely many solutions. • Independent System – a system of equations that has exactly one solution. • Inconsistent System – a system of equations or inequalities that has no solution.