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**Unformatted text preview: **We use matrices to model systems of linear equations, both homogeneous and inhomogeneous. 2x − x+ 2x − x+ 3y y 3y y + − +z − 5z z 5z = = = = 7 3 0 0 2 1 −3 1 1 −5 17 −5 3 2 −3 11 Rather than work with systems of equations, we prefer to work with the the corresponding matrices. Theorem. Any matrix M can be transformed into a matrix M in echelon form by a sequence of elementary row operations. The system of equations corresponding to M will be equivalent to (have exactly the same solutions as) the system of equations corresponding to M . The echelon form of 2 −3 11 1 −5 is 1 0 0 1 −14/15 −11/5 . The homogeneous system of equations on the preceding page is equivalent to x − (14/15)z = 0 y − (11/5)z = 0 The echelon form of 2 1 −3 1 17 −5 3 is 1 0 0 −14/15 16/5 1 −11/5 −1/5 . The homogeneous system of equations on the preceding page is equivalent to x = 16/5 + (14/15)z y = −1/5 + (11/5)z It is convenient to combine some elemenary row operations into a single operation we call a pivot operation. If a11 ⎜ a21 M =⎜ . ⎝. . am1 ⎛ a12 a22 . . . am2 ... ... ⎞ aan a2n ⎟ .⎟ .⎠ . ... . . . amn and aij = 0, we may pivot on position (i, j ) by, ﬁrst, multiplying row i by 1/aij so that the (i, j ) position becomes 1, and then adding (or subtracting) multiples of row i from the other rows so that all entries of column j , except the entry in position (i, j ) become 0. For example, if we pivot on position (1, 1) of a 3 by 2 matrix, we get ⎛ ⎞ 1 a21 /a11 ⎝ 0 a22 − a21 a12 /a11 ⎠ 0 a32 − a31 a12 /a11 ...

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