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10.3 Systems of Linear Equations: Matrices. A matrix is defined as a rectangular array of numbers,. Column j. Column n. Column 1. Column 2. Row 1. Row 2. Row 3. Row 4. Augmented Matrix:. Row Operations on an Augmented Matrix. 1. Interchange any two rows.

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## 10.3 Systems of Linear Equations: Matrices

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**A matrix is defined as a rectangular array of numbers,**Column j Column n Column 1 Column 2 Row 1 Row 2 Row 3 Row 4**Row Operations on an Augmented Matrix**1. Interchange any two rows. 2. Replace a row by a nonzero multiple of that row. 3. Replace a row by the sum of that row and a constant multiple of some other row.**Solve**Find the augmented matrix: Find the echelon form. R2=-2 R1+R2**R2 =R2/3**R3 =-4R2+R3**R3=R3*(-3/25)**The third row of the matrix represents the equation z =-7/25. Substituting this into the equation represented by the second row we get:**Let z =-7/25, y =-44/25 in the first:**Solution is:**Solve**using a graphing utility.**Substitute z = 5 into the second.**Substitute z=5, y =-2 into the first.**Solve using a graphing utility:**Using rref(.) function we get: Dependent system: Infinitely many solutions.**Solve for y from the second:**Solve for x from the first: Solution is (x, y, z) = (18/5 - (7/5)z, 7/5 + (2/5)z, z)

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