Linear Algebra

To find the LU decomposition of the matrix A, we will perform Gaussian elimination to transform A into an upper triangular matrix U, while keeping track of the elementary row operations through a lower triangular matrix L. If necessary, we will use partial pivoting and keep track of the permutation matrix P. The matrix A is given by: $$ A = \begin{bmatrix} 4 & 2 & -1 & -1 \\ -16 & -10 & 6 & 4 \\ 0 & 6 & -7 & 1 \\ -4 & -2 & 1 & 0 \end{bmatrix} $$ Let's start the process of Gaussian elimination. Step 1: We look at the first column and see that the pivot element, 4, is not the largest in magnitude. The largest in magnitude is -16. So we will swap the first and second rows and keep track of this swap with a permutation matrix P. $$ P = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ After swapping, the matrix A becomes: $$ A = \begin{bmatrix} -16 & -10 & 6 & 4 \\ 4 & 2 & -1 & -1 \\ 0 & 6 & -7 & 1 \\ -4 & -2 & 1 & 0 \end{bmatrix} $$ Step 2: Now we will eliminate the elements below the pivot in the first column by adding multiples of the first row to the rows below. To eliminate the 4 in the second row, we add 1/4 of the first row to the second row. To eliminate the -4 in the fourth row, we add 1/4 of the first row to the fourth row. The corresponding row operations give us the following matrix: $$ A = \begin{bmatrix} -16 & -10 & 6 & 4 \\ 0 & 0.5 & -2.5 & -2 \\ 0 & 6 & -7 & 1 \\ 0 & -0.5 & 2.5 & 1 \end{bmatrix} $$ And the L matrix starts to take shape: $$ L = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1/4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1/4 & 0 & 0 & 1 \end{bmatrix} $$ Step 3: Next, we will focus on the second column and use the second row as the pivot row. We see that the pivot element, 0.5, is not the largest in magnitude in its column. The largest in magnitude is 6. So we will swap the second and third rows and update the permutation matrix P. $$ P = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ After swapping, the matrix A becomes: $$ A = \begin{bmatrix} -16 & -10 & 6 & 4 \\ 0 & 6 & -7 & 1 \\ 0 & 0.5 & -2.5 & -2 \\ 0 & -0.5 & 2.5 & 1 \end{bmatrix} $$ Step 4: Now we will eliminate the 0.5 and -0.5 in the third and fourth rows of the second column by adding multiples of the second row to these rows. To eliminate the 0.5 in the third row, we add -1/12 of the second row to the third row. To eliminate the -0.5 in the fourth row, we add 1/12 of the second row to the fourth row. The corresponding row operations give us the following matrix: $$ A = \begin{bmatrix} -16 & -10 & 6 & 4 \\ 0 & 6 & -7 & 1 \\ 0 & 0 & -2 & -2 \\ 0 & 0 & 3 & 1 \end{bmatrix} $$ And we update the L matrix: $$ L = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1/4 & 1 & 0 & 0 \\ 0 & -1/12 & 1 & 0 \\ 1/4 & 1/12 & 0 & 1 \end{bmatrix} $$ Step 5: Finally, we will focus on the third column and use the third row as the pivot row. We don't need to swap rows since the pivot element, -2, is the only non-zero element in its column. We will eliminate the 3 in the fourth row by adding 3/2 of the third row to the fourth row. The corresponding row operation gives us the following matrix: $$ A = \begin{bmatrix} -16 & -10 & 6 & 4 \\ 0 & 6 & -7 & 1 \\ 0 & 0 & -2 & -2 \\ 0 & 0 & 0 & 4 \end{bmatrix} $$ And we update the L matrix: $$ L = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1/4 & 1 & 0 & 0 \\ 0 & -1/12 & 1 & 0 \\ 1/4 & 1/12 & 3/2 & 1 \end{bmatrix} $$ Now we have our upper triangular matrix U and our lower triangular matrix L, along with the permutation matrix P. The PA=LU decomposition is: $$ P = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad L = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1/4 & 1 & 0 & 0 \\ 0 & -1/12 & 1 & 0 \\ 1/4 & 1/12 & 3/2 & 1 \end{bmatrix}, \quad U = \begin{bmatrix} -16 & -10 & 6 & 4 \\ 0 & 6 & -7 & 1 \\ 0 & 0 & -2 & -2 \\ 0 & 0 & 0 & 4 \end{bmatrix} $$ This completes the PA=LU decomposition of the given matrix A.

Math

Find either A=LU or PA=LU decomposition of the following matrix: A = | 4 2 -1 -1 | | -16 -10 6 4 | | 0 6 -7 1 | | -4 -2 1 0 |

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