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8.2 OPERATIONS WITH MATRICES Copyright © Cengage Learning. All rights reserved. What You Should Learn • Decide whether two matrices are equal. • Add and subtract matrices and multiply matrices by scalars. • Multiply two matrices. • Use matrix operations to model and solve real-life problems. 2 Equality of Matrices 3 Equality of Matrices 4 Equality of Matrices Two matrices A = [aij] and B = [bij] are equal if they have the same order (m n) and aij = bij for 1 i m and 1 j n. In other words, two matrices are equal if their corresponding entries are equal. 5 Example 1 – Equality of Matrices Solve for a11, a12, a21, and a22 in the following matrix equation. Solution: a11 = 2, a12 = –1, a21 = –3, and a22 = 0. 6 Matrix Addition and Scalar Multiplication 7 Matrix Addition and Scalar Multiplication Matrix Subtraction A – B = A + (-B) 8 Example 2 – Addition of Matrices 9 Example 2 – Addition of Matrices cont’d d. The sum of and is undefined because A is of order 3 3 and B is of order 3 2. 10 Matrix Addition and Scalar Multiplication 11 Matrix Addition and Scalar Multiplication If A is an m n matrix and O is the m n zero matrix consisting entirely of zeros, then A + O = A. O is the additive identity for the set of all m n matrices. 2 3 zero matrix 2 2 zero matrix 12 Matrix Addition and Scalar Multiplication Real Numbers (Solve for x.) m n Matrices (Solve for X.) x + a= b X+A=B x + a + (–a) = b + (–a) X + A + (–A) = B + (–A) x+0 =b–a X+O=B–A x=b–a X=B–A 13 Matrix Multiplication 14 Matrix Multiplication For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. A B = AB mn np mp 15 Example 7 – Finding the Product of Two Matrices Find the product AB using Solution: and 16 Matrix Multiplication For most matrices, AB BA. 17 Matrix Multiplication 18 Matrix Multiplication If A is an n n matrix, the identity matrix has the property that AIn = A and InA = A. AI = A and IA = A 19 Applications 20 Applications A X = B 21 Example 11 – Solving a System of Linear Equations Consider the following system of linear equations. x1 – 2x2 + x3 = – 4 x2 + 2x3 = 4 2x1 + 3x2 – x3 = 2 a. Write this system as a matrix equation, AX = B. b. Use Gauss-Jordan elimination on the augmented matrix to solve for the matrix X. 22 Example 11 – Solution a. In matrix form, AX = B, the system can be written as follows. b. The augmented matrix is formed by adjoining matrix B to matrix A. 23 Example 11 – Solution cont’d Using Gauss-Jordan elimination, you can rewrite this equation as So, the solution of the system of linear equations is x1 = –1, x2 = 2, and x3 = 1, and the solution of the matrix equation is 24