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Type I row operations: Replace row i by row i plus a multiple (say λ ) of row j . We write this as Ri↦Ri+λRj . In terms of equations, this means we're adding/subtracting a multiple of equation j to equation i .
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Type II row operations: Replace row i by a nonzero multiple (say λ≠0 ) of row i . We write this as Ri↦λRi . In terms of equations, this means we're multiplying an equation by a nonzero constant.
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Type III row operations: Swap row i and row j . This corresponds to reordering your equations.
13. Simultaneous equations and row operations
13. Simultaneous equations and row operations
Thus far in the course, we have focused on the geometric aspects of matrices and the transformations they determine. Now we'll approach the subject from the point of view of simultaneous equations.
Simultaneous equations
A system of simultaneous linear equations, for example x-y=-1,x+y=3,
A system of simultaneous linear equations is a matrix equation in disguise. For example, the system above can be written as (1-111)(xy)=(-13).
We will abbreviate such a matrix equation by writing a so-called augmented matrix: we write the matrix of coefficients, then a vertical bar, then the column of constants: (1-1|-111|3)
Solving these equations
To solve this system, we will manipulate the equations one at a time. We will see what happens to the augmented matrix as we perform these manipulations. We start with: x-y=-1,x+y=3,(1-1|-111|3).
Row operations
This process of solving simultaneous equations can therefore be understood as performing a sequence of row operations on the augmented matrix.
We don't allow ourselves to multiply an equation by zero: this will change our system of equations by effectively ignoring some of them.
In our example, we "solved" the equation when we reached x=1,y=2 . This meant that the augmented matrix had the identity matrix on the left of the vertical bar. So the aim of the row operations is to put the augmented matrix in the form (I|b) where I is the identity matrix and b is a column vector (of "constants"). Of course, this will sometimes fail:
Consider the system x+y=1 (one equation, two variables). The augmented matrix is now (11|1).