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Math 1390 - Manyo 4.2 - page 1 of 3 4.2 – Systems of Linear Equations and Augmented Matrices Read pages 188 - 197 Homework: page196 Matrices: 1, 3, 5, 7, 9, 11, 13, 15, 21, 25, 27 Solving Linear Systems Using Augmented Matrices: 35, 41, 45, 47, 49, 53, 59, 61, 63 Q1: Use this matrix to learn the terminology and answer the questions 1 A 2 3 4 5 6 1. If a matrix has m rows and n columns, it is called an m n matrix. The numbers m and n are called the dimensions of the matrix. What are the dimensions of A ? _____________________ 2. Each number in a matrix is called an element of the matrix. The position of each element, a i j , is given using double subscript notation by its row number, i , and its column number, j . Using the matrix A, what are a 2 1 = ______ a1 3 = _______ Rewrite (above) the matrix A using the a i j 3. A matrix with the same number of rows as columns (m = n), is called a square matrix of order n. 4. A matrix with one column is called a column matrix and has dimension m x 1 5. A matrix with one row is called a row matrix and has dimension 1 x n Classify each matrix and give its dimension B 1 2 3 __________________________ 3 C 2 3 5 4 D 3 2 4 5 4 8 6 9 __________________________ _______________________ 6. The elements a11, a22 , a33 ,.... make up the elements in the principal diagonal of a matrix. List the elements in the principal diagonal of the matrices A and C . 7. The augmented matrix of the system of linear equations a11x1 a12 x2 k1 a21x1 a22 x2 k 2 a11 a12 k1 a 21 a 22 k 2 is Math 1390 - Manyo 4.2 - page 2 of 3 Q2: Set up the augmented matrix for the system x1 3x2 2 4 x1 6 x2 8 An augmented matrix is transformed into a row – equivalent matrix by performing any of the following row operations: Two rows are interchanged R i Rj : Ri and R j are interchanged A row is replaced by a nonzero, constant multiple of itself cRi Ri : cRi replaces Ri A constant multiple of one row is added to another row cR j Ri Ri : cR j Ri replaces Ri Q2: Perform the following operations on the augmented matrix from the previous page. R1 R2 1 R2 R2 4 (4) R1 R2 R2 1 3 2 4 6 8 1 3 2 4 6 8 1 3 2 4 6 8 Q3: Solve each augmented matrix 1 0 2 0 3 - 6 A. 1 2 4 0 1 1 B. Math 1390 - Manyo 4.2 - page 3 of 3 Q4: In the left column, solve the systems using augmented matrices. To the right is a summary of the possible final matrix forms for a linear system of two Equations in two variables. Note: m, n and p are real numbers and p 0 A. −3𝑥1 − 𝑥2 = 5 𝑥1 − 3𝑥2 = −5 Form 1: A unique solution Consistent and Independent 1 0 m 0 1 n 𝑥1 = 𝑚 𝑥2 = 𝑛 B. 𝑥1 − 2𝑥2 = −3 −2𝑥1 + 4 𝑥2 = 6 Form 2: Infinitely many solutions Consistent and Dependent 1 m n 0 0 0 𝑥1 = 𝑛 − 𝑚𝑘 𝑥2 = 𝑘 C. 𝑥1 − 2𝑥2 = −3 −2𝑥1 + 4 𝑥2 = 8 Form 3: No solution Inconsistent 1 m n 0 0 p