Determine the column of the matrix A are linearly independent, where

\(A = \begin{bmatrix}0 & 1 & 4\\ 1 & 2 & -1\\ 5 & 8 & 0\end{bmatrix}\)

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Solution:

Given matrix is

\(A = \begin{bmatrix}0 & 1 & 4\\ 1 & 2 & -1\\ 5 & 8 & 0\end{bmatrix}\)

To check it independency, We have an augmented form

\(\begin{bmatrix}0 & 1 & 4 & : & 0\\ 1 & 2 & -1 & : & 0\\ 5 & 8 & 0 & : & 0\end{bmatrix}\)

R1 ↔ R3

~ \(\begin{bmatrix}1 & 2 & -1 & : & 0\\ 0 & 1 & 4 & : & 0\\ 5 & 8 & 0 & : & 0\end{bmatrix}\)

R3 → R3 – 5R1

~ \(\begin{bmatrix}1 & 2 & -1 & : & 0\\ 0 & 1 & 4 & : & 0\\ 0 & -2 & 5 & : & 0\end{bmatrix}\)

R1 → R1 – 2R2 ,  R3 → R3 + 2R1

~ \(\begin{bmatrix}1 & 0 & -9 & : & 0\\ 0 & 1 & 4 & : & 0\\ 0 & 0 & 3 & : & 0\end{bmatrix}\)

R3 → R3 / 3

~ \(\begin{bmatrix}1 & 0 & -9 & : & 0\\ 0 & 1 & 4 & : & 0\\ 0 & 0 & 1 & : & 0\end{bmatrix}\)

R2 → R2 – 4R3,  R1 → R1 + 9R3

~ \(\begin{bmatrix}1 & 0 & 0 & : & 0\\ 0 & 1 & 0 & : & 0\\ 0 & 0 & 1 & : & 0\end{bmatrix}\)

Hence, It is linearly independent.

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