ss_10_3 - Section 10.3 Matrices Matrices have many uses...

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Section 10.3 Matrices Matrices have many uses, most of which will be discussed in later sections. In this section, we will use matrices to help us solve systems of linear equations. A matrix is simply a rectangular array of numbers. Several notations are used for matrices. We will use only the following notation. Examples of matrices: 1 2 - 3 - 1 - 3 - 1 2 2 1 2 3 3 2 1 2 1 0 4 - 1 2 3 3 - 1 4 - 2 The following is common notation for a matrix with m rows and n columns. A matrix with m rows and n columns is said to be of order ”m by n”, often written ”m x n”: a 11 a 12 · · · a 1 j · · · a 1 n a 21 a 22 · · · a 2 j · · · a 2 n · · · a i 1 a i 2 · · · a ij · · · a in · · · a m 1 a m 2 · · · a mj · · · a mn a ij is the entry in the i-th row and j-th column of the matrix A Matrices associated with linear systems Two types of matrices are commonly used for solving linear systems. They are coefficient matrices and augmented matrices . These two types of matrices are shown in the following example. Example. The coefficient matrix, A , and the augmented matrix, B , for the following system are shown below. 2 x - y + z = 3 x - y = 2 y + 2 z = - 3 A = 2 - 1 1 1 - 1 0 0 1 2 Coefficient matrix B = 2 - 1 1 | 3 1 - 1 0 | 2 0 1 2 | - 3 Augmented matrix Example. Solve the following system using matrices. 2 x - y + z = 3 x - y = 2 y + 2 z = - 3 Solution. We begin with the augmented matrix for the system. Then we use elementary row operations to simplify the augmented matrix to one having zeros below the main diagonal of the matrix. The simplified matrix is the matrix of a system equivalent to the original system.
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