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12.2 SystemsLinearEqMatrices

12.2 SystemsLinearEqMatrices - bar appear at the bottom If...

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12.2 Systems of Linear Equations: Matrices A matrix is a rectangular array of numbers. Each number a ij of the matrix has to indices (subscripts), the row index i and the column index j (see your book for an illustration). A matrix can be used to represent a system of linear equations. If the constants to the right of the equal sign are included, it is called an augmented matrix ; if the constants are not included, it is called a coefficient matrix . To solve a system, there are three basic row operations : [1] Interchange any two rows; [2] replace a row by a nonzero multiple of that row; and [3] replace a row by the sum of that row and a constant nonzero multiple of another row. The goal is to get the matrix in row echelon form , which means [1] entry a 11 is a 1 and only zeros appear below it; [2] the first nonzero entry in each subsequent row is a 1 with zeros below it and it is to the right of the first nonzero entry in any row above it; and [3] any row that contains all zeros to the left of the vertical
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Unformatted text preview: bar appear at the bottom. If the bottom row is all zeros, both left and right of the vertical bar, the system is dependent, meaning infinitely man solutions. If the bottom row has zeros to the left of the vertical bar and nonzero number on the right, it is inconsistent, meaning no solution. Solve the system using row operations to get the augmented matrix in row echelon form: [1] Augmented matrix [2] a 11 = 1 [3] a 21 = 0 1. 5 3 4 4 10 x y x y +-= -= 2. 4 3 4 1 6 4 2 5 2 3 22 x y z x y x y z--= -+ = + + = 3. 2 3 3 2 2 2 5 3 2 x y z x y z x y z-- = + + = + + = Try These: 4. 4 5 3 2 5 2 5 4 3 13 2 2 2 2 w x w x y w x y z w x y z-= -+-= - -+ + = +-- = - 5. 2 3 2 2 6 3 3 5 x y z x y z x y z +- = -+ = -+ =...
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