Question

For each of the following matrices, determine whether the columns of the matrix are linearly independent or linearly dependent. If they dependent, give a non-zero vector x such that Ax-0.

a) A= [5 10 5 ]
[-3 -3 6]
[1 4 8]
the columns of A are linearly dependent
x=[0 ]
[0]
[0]

Answer 1

The columns of the matrix A are linearly independent.

Step-by-step explanation:

The given matrix A is:

A = [5 10 5]
[-3 -3 6]
[1 4 8]

To determine whether the columns of A are linearly independent or linearly dependent, we need to check if there exist scalars (non-zero vector x) such that Ax = 0.

We can find the determinant of matrix A.

If the determinant is zero, then the columns of A are linearly dependent. Otherwise, they are linearly independent.

Let's calculate the determinant:

|A| = 5(-3)(8) + 10(6)(1) + 5(-3)(4) - 1(-3)(5) - 4(6)(5) - 8(-3)(1)

|A| = -120 + 60 + (-60) - (-15) - 120 - (-24)

|A| = -120 + 60 - 60 + 15 -120 + 24

|A| = -201

Since the determinant is nonzero (-201 != 0), the columns of A are linearly independent.