Question

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix.[1 −3 −8 5 0 1 2 −4]

Answer 1

The given matrix [1 -3 -8 5 0 1 2 -4] is row equivalent to the augmented matrix [1 -3 -8 | 5], which is in reduced row echelon form. From this, we can see that the system of linear equations Ax = 0 has three linearly independent columns, corresponding to the variables x1, x2, and x4. The other variables, x3 and x5, are dependent on x1, x2, and x4 and can be expressed in terms of them.

Using the values from the reduced row echelon form, we can write:

x3 = -5/8 * x1 - 3/8 * x2
x5 = 5/2 * x1 + 1/2 * x2 + 4 * x4

So, all solutions of Ax = 0 can be expressed in terms of x1, x2, and x4, using the above equations:

x = x1 * [1, 0, -5/8, 0, 5/2, 4 * x4]