\subsection{Problem 10}
Transform  the  following  matrix  to  the  RREF,  determine $rank(\mathbf{A})$ and  identify  the columns corresponding to the basic and free variables. 
\begin{equation*}
    \matr{A} = 
    \begin{bmatrix}
    1 & 2 & 2 & 3 & 1 \\
    2 & 4 & 4 & 6 & 2 \\
    3 & 6 & 6 & 9 & 6 \\
    1 & 2 & 4 & 5 & 3 
    \end{bmatrix}
\end{equation*}

Check whether the symmetric matrix $\mathbf{A}$ is positive-definite. If so, apply the Cholesky factorization. Then, compute its inverse.

\subsubsection*{Solution}

\begin{equation*}
    \begin{split}
    \begin{bmatrix}
    1 & 2 & 2 & 3 & 1 \\
    2 & 4 & 4 & 6 & 2 \\
    3 & 6 & 6 & 9 & 6 \\
    1 & 2 & 4 & 5 & 3 
    \end{bmatrix}
    \xrightarrow{\substack{R_2 - 2R_1 \\ R_3 - 3R_1 \\ R_4 - R_1}}
    \begin{bmatrix}
    1 & 2 & 2 & 3 & 1 \\
    0 & 0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 & 3 \\
    0 & 0 & 2 & 2 & 2
    \end{bmatrix}
    \xrightarrow{pivot}
    \begin{bmatrix}
    1 & 2 & 2 & 3 & 1 \\
    0 & 0 & 2 & 2 & 2 \\
    0 & 0 & 0 & 0 & 3 \\
    0 & 0 & 0 & 0 & 0
    \end{bmatrix}
    \\
    \xrightarrow{\substack{R_2 / 2 \\ R_3 / 3}}
    \begin{bmatrix}
    1 & 2 & 2 & 3 & 1 \\
    0 & 0 & 1 & 1 & 1 \\
    0 & 0 & 0 & 0 & 1 \\
    0 & 0 & 0 & 0 & 0
    \end{bmatrix}
    \xrightarrow{\substack{R_1 - R_3 \\ R_2 - R_3}}
    \begin{bmatrix}
    1 & 2 & 2 & 3 & 0 \\
    0 & 0 & 1 & 1 & 0 \\
    0 & 0 & 0 & 0 & 1 \\
    0 & 0 & 0 & 0 & 0
    \end{bmatrix}
    \end{split}
\end{equation*}

Rank of the given matrix is equal to 3.