Use the procedure illustrated in Examples 3 and 4 to find the LU-factorization of the given matr

Use the procedure illustrated in Examples 3 and 4 to find...

Key Concepts

Elementary Elimination Matrices

Elementary elimination matrices represent the individual row operations performed during Gaussian elimination. By capturing these operations, they help in constructing the lower triangular matrix L, as each multiplier used in the elimination is recorded and can be transferred into L to complete the LU factorization.

Upper Triangular Matrix

An upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero. This structure is produced after applying Gaussian elimination, and in the context of LU factorization, it constitutes the U matrix, containing the resulting coefficients after the elimination process.

Gaussian Elimination

Gaussian elimination is the process of applying a series of row operations to transform a matrix into an upper triangular form. This method systematically eliminates the entries below the pivot positions and is the procedural basis for obtaining the U matrix in LU factorization.

LU Factorization

LU factorization is a method of decomposing a square matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U). This decomposition is fundamental in solving systems of linear equations, computing determinants, and inverting matrices because it simplifies these operations by reducing them to forward and backward substitution steps.

Lower Triangular Matrix

A lower triangular matrix is a square matrix in which all the entries above the main diagonal are zero. In LU factorization, it appears as the matrix L, where the entries below the main diagonal are typically the multipliers used to eliminate terms during Gaussian elimination, and the diagonal entries are usually set to one.

Transcript

00:01 In this question, we need to find the allu factorization of the given matrix, which is given as a matrix a, having, which is of order 3 by 3, having the elements 4, 2, 12, 2, 26 minus 4 and 56.

00:16 So we need to use the method which is illustrated in the examples 3 and 4.

00:21 So first of all, we will find out the upper triangular matrix that is u by reducing these three elements to the two elements to the two.

00:30 To zero using row operations.

00:31 So this one, this one and this one, that is a21, a31, and a32, these elements to be converted to zero by using row operations.

00:40 So we will use the row operation that is r2 stores, r2 minus half of r1.

00:51 And we will record the negative of the multiple of r1 here, that is negative of minus half, which will be half.

00:58 And this matrix will be changed as, that is, matrix having the elements 4 to 12 to 26 and minus 4 12 minus 4 and 56 this will be changed as here 4 to 12 and now r2 to be changed as r2 minus half of r1 so 2 minus half of 4 that will be 2 minus 2 which is 0 here 26 minus half of 2 that will be 26 minus 1 that will be 25 and minus 4 minus half of 12 that will be minus 4 minus 6 that will become minus 10 and row 3 to be remain same 12 minus 4 and 56 now from here we will record the negative multiple of r 1 that is half so record half this half to be recorded as for this one element that is a 2 1 and now we will we will change that we will use the row operation for changing this one element that is a 3 1 which is 12 here so we will use a row operation to change this to 0 we will use the operation as r3 stores r3 minus thrice of our 1 so we will use this row operation and this will become that is 4 to 12 025 minus 10 and 12 minus 4 56 will become this will be changed as that is 4 to 12 0 25 minus 10 this will become 12 minus 3 times 4 as r3 minus 3 times r1 this will become 12 minus 12 which is 0 and you have minus 4 minus 3 times 2 that will be minus 4 minus 6 that will become that is minus 10 and you have 56 minus 3 times 12 that will be 56 minus 36 and that will be 20 now we need to make this element 0 so we will add that is we will change in r3 and first of all we will record from here the negative multiple of that is r1 that is that will be negative of minus 3 so record 3 here record 3 and this is for the element that is a 3 1 this one element so now we will use again a row operation to reduce this minus 10 to 0 so we will use the row operation that is r3 stores r3 minus 2 by fifth of r2 sorry not minus this will be plus 2 by fifth of r2 so from here this matrix as we have changed already that is 4 to 12 025 and minus 10 the 0 minus 10 20 this will be changed to that is 4 to 12 0 25 minus 10 and here 0 minus 2 by 5th of 0 will be 0 minus 10 minus plus 2 by 5th of 25 that will become minus 10 plus 10 which will be 0 and 20 plus 2 by 5th of minus 10 that will be 20 and minus 4 that will be 16 okay so from here we will record the negative of the multiple of r2 so we will record here minus 2 by 5 and this is recorded for this one element that is a 3 2 now we will so this matrix we have here found is actually the upper triangle matrix this so this one is the u this one is the u matrix now we will find out the l matrix that is lower triangular matrix by using a identity matrix of order 3 by 3 and changing the the elements that is by according to the recorded values so the identity matrix that is 3 by 3 matrix of order 3 3 3 3 3 so this will be 1 0 0 0 0 0 0 0 0 0 0 1 this will be changed according to the recorded values as per the the elements that is 821 a 3 1 and 8 3 2 2 so this will become 8 for the recorded elements were that is half 3 and minus 2 by 5 so here this will be half and this will be 1 0 and here this was 3 and this one was minus 2 by 5 so here this will become minus 2 by 5 and and this one was one.

06:54 So this matrix is actually lower triangular matrix which is l.

06:59 So we will we can write it as l here.

07:03 So this one is l now we can write the matrix a as the product of l and u so we will write a as l product of l and u as lu...

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