Question : Apply LU decomposition method to solve 2x+4y+2z=1,x+5y+2z=1,4x-y+9z=2Answer : Writing system in the Matrix AX= b\[\left[\begin{matrix}2&4&2\\1&5&2\\4&-1&9\\\end{matrix}\right]\left(\begin{matrix}x\\y\\z\\\end{matrix}\right)=\left(\begin{matrix}1\\1\\2\\\end{matrix}\right)\]To find LU decomposition of A\[A=\left[\begin{matrix}2&4&2\\1&5&2\\4&-1&9\\\end{matrix}\right]\]Step 1A=LUWhere L is lower triangular matrix and U is upper triangular matrix respectively\[L=\left[\begin{matrix}l_{11}&0&0\\l_{21}&l_{22}&0\\l_{31}&l_{32}&l_{33}\\\end{matrix}\right],U=\left[\begin{matrix}u_{11}&u_{12}&u_{13}\\0&u_{22}&u_{23}\\0&0&u_{33}\\\end{matrix}\right]\]Therefore \[A=LU=\left[\begin{matrix}l_{11}&0&0\\l_{21}&l_{22}&0\\l_{31}&l_{32}&l_{33}\\\end{matrix}\right],\left[\begin{matrix}u_{11}&u_{12}&u_{13}\\0&u_{22}&u_{23}\\0&0&u_{33}\\\end{matrix}\right]\]\[\left[\begin{matrix}2&4&2\\1&5&2\\4&-1&9\\\end{matrix}\right]=\left[\begin{matrix}u_{11}&u_{12}&u_{13}\\l_{21}u_{11}&{l_{21}u_{12}+u}_{22}&l_{21}u_{13}+u_{23}\\l_{31}u_{11}&l_{31}u_{12}+l_{32}u_{22}&{l_{31}u_{13}+l_{32}u_{23}+u}_{33}\\\end{matrix}\right]\]\[u_{11}=2,\ \ u_{12}=4,\ \ \…
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