59 lines
1.5 KiB
TeX
59 lines
1.5 KiB
TeX
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\subsection{Problem 10}
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Transform the following matrix to the RREF, determine $rank(\mathbf{A})$ and identify the columns corresponding to the basic and free variables.
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\begin{equation*}
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\matr{A} =
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\begin{bmatrix}
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1 & 2 & 2 & 3 & 1 \\
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2 & 4 & 4 & 6 & 2 \\
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3 & 6 & 6 & 9 & 6 \\
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1 & 2 & 4 & 5 & 3
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\end{bmatrix}
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\end{equation*}
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Check whether the symmetric matrix $\mathbf{A}$ is positive-definite. If so, apply the Cholesky factorization. Then, compute its inverse.
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\subsubsection*{Solution}
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\begin{equation*}
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\begin{split}
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\begin{bmatrix}
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1 & 2 & 2 & 3 & 1 \\
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2 & 4 & 4 & 6 & 2 \\
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3 & 6 & 6 & 9 & 6 \\
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1 & 2 & 4 & 5 & 3
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\end{bmatrix}
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\xrightarrow{\substack{R_2 - 2R_1 \\ R_3 - 3R_1 \\ R_4 - R_1}}
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\begin{bmatrix}
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1 & 2 & 2 & 3 & 1 \\
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0 & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 3 \\
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0 & 0 & 2 & 2 & 2
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\end{bmatrix}
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\xrightarrow{pivot}
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\begin{bmatrix}
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1 & 2 & 2 & 3 & 1 \\
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0 & 0 & 2 & 2 & 2 \\
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0 & 0 & 0 & 0 & 3 \\
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0 & 0 & 0 & 0 & 0
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\end{bmatrix}
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\\
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\xrightarrow{\substack{R_2 / 2 \\ R_3 / 3}}
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\begin{bmatrix}
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1 & 2 & 2 & 3 & 1 \\
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0 & 0 & 1 & 1 & 1 \\
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0 & 0 & 0 & 0 & 1 \\
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0 & 0 & 0 & 0 & 0
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\end{bmatrix}
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\xrightarrow{\substack{R_1 - R_3 \\ R_2 - R_3}}
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\begin{bmatrix}
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1 & 2 & 2 & 3 & 0 \\
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0 & 0 & 1 & 1 & 0 \\
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0 & 0 & 0 & 0 & 1 \\
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0 & 0 & 0 & 0 & 0
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\end{bmatrix}
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\end{split}
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\end{equation*}
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Rank of the given matrix is equal to 3.
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