770 CHAPTER 11 Systems of Equations and Inequalities In Words To augment means to increase or expand. An augmented matrix broadens the idea of matrices to systems of linear equations. CAUTION Be sure variables and constants are lined up correctly before writing the augmented matrix. j Writing the Augmented Matrix of a System of Linear Equations Write the augmented matrix of each system of equations. (a) − =− − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ (1) (2) x y x y 3 4 6 2 3 5 (b) x y z x z x y 2 0 1 0 2 8 0 (1) (2) (3) − + = + − = + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ EXAMPLE 1 Solution (a) The augmented matrix is − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 3 2 4 3 6 5 (b) Care must be taken that the system be written so that the coefficients of all variables are present (if any variable is missing, its coefficient is 0). Also, all constants must be to the right of the equal sign. We need to rearrange the given system to put it into the required form. − + = + − = + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) x y z x z x y 2 0 1 0 2 8 0 − + = + ⋅ + = + + ⋅ = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) x y z x y z x y z 2 0 0 1 2 0 8 The augmented matrix is ⎡ − ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 2 1 1 1 0 2 1 1 0 0 1 8 on the right side of the equal signs. The vertical bar serves as a reminder of the equal signs. The large square brackets are used to denote a matrix in algebra. DEFINITION Matrix A matrix is defined as a rectangular array of numbers: ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ j n m Column 1 Column 2 Column Column Row 1 Row 2 Row Row a a a a a a a a a a a a a a a a i j n j n i i ij in m m mj mn 11 12 1 1 21 22 2 2 1 2 1 2 (1) Each number aij in the matrix has two indexes: the row index i and the column index j. The matrix shown in display (1) has m rows and n columns. The numbers aij are usually referred to as the entries of the matrix. For example, a23 refers to the entry in the second row, third column. 1 Write the Augmented Matrix of a System of Linear Equations Now we will use matrix notation to represent a system of linear equations.The matrix used to represent a system of linear equations is called an augmented matrix . In writing the augmented matrix of a system, the variables of each equation must be on the left side of the equal sign and the constants on the right side.A variable that does not appear in an equation has a coefficient of 0.
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