Matrices

The example in figure 1 which was patterned after an example in Smith (2003 ) is an example of a matrix with no solutions . According to Smith a system of equations that has no solutions would form a matrix whose reduced form obtained through a series of elementary row operations contains at least one row with all elements equal to zero except for the element of the last column in that row . If we perform an elementary row operation on the matrix in figure 1 as demonstrated in figure 2 , we can see that the

resulting matrix would have the bottom row as Smith described it . An explanation for this derived from Strang (1998 ) is that the bottom row represents the equation 0 1 which cannot be . Thus , the matrix cannot have a solution

Figure 1 1 2 4 3 4 2 3 1 8 4 6 3

Figure 2 1 2 0 3 4 2 3 1 0 0 0 1 Operation : -2row2 row3 On the other hand , matrices that contain rows that are entirely consisting of zeroes can have many solutions (Smith , 2003 . The matrix in figure 3 which is a modification from the one in figure 1 is one such example . When we perform the same elementary row operation in figure 4 as we did in figure 2 , we observe that the last row is now entirely of zeroes . This can have many solutions as one of the constraints was removed leaving only two equations in three unknowns , a situation which is known to have many solutions . Hence , a unique solution cannot be provided

Figure 3 1 2 4 3 4 2 3 1 8 4 6 2

Figure 4 1 2 0 3 4 2 3 1 0 0 0 0 Operation : -2row2 row3

References

Smith , L (2003 . Linear Algebra 3rd Edition . Prentice Hall : N .Y

Strang , G (1998 . Introduction to Linear Algebra . Wellesley-Cambridge Press : M .A...