Thread: Gaussian Elimination to ensure solution

I have Q, a 5 x 4 matrix and I need to find the conditions on b_1,b_2,b_3,b_4,b_5 that ensure that the system Qx=b has a solution by choosing an appropriate (A|b) and using Gaussian Elimination

how would I go about doing this?

should my a and b be

Q and b_1,b_2,b_3,b_4,b_5 respectively?

many thanks

Originally Posted by James0502

I have Q, a 5 x 4 matrix and I need to find the conditions on b_1,b_2,b_3,b_4,b_5 that ensure that the system Qx=b has a solution by choosing an appropriate (A|b) and using Gaussian Elimination

how would I go about doing this?

should my a and b be

Q and b_1,b_2,b_3,b_4,b_5 respectively?

many thanks

Yes, that is correct. Form the "augmented" matrix consisting of Q with b1, b2, b3, b4, b5 forming the 5th column. Row reduce. You will always be able to reduce the first 4 number on the last row to 0. (And, if the equations are not independent, perhaps on higher rows.) The last column will be, of course, combinations of the "b"s. The equation will have a solution if and only if the numbers in the last column for "all 0 rows" are also 0. Those will be be conditions on the "b"s.

Ok.. I end up with an upper triangular matrix, with the last column made up of multiples of b, as you said.. does this mean it has a solution.. even though the last row is made up of zeros, with b's in the last column?

many thanks