Prove directly that the matrix A=( 0 1 1 0 ) does not have an L U factorizatio

step 1 Hello students, we are given a matrix here. We need to show and we need to show that this matrix

Prove directly that the matrix A=( 0 1 1 ...

Key Concepts

LU Factorization

LU factorization is the process of decomposing a given square matrix into the product of a lower triangular matrix (typically with ones on the diagonal) and an upper triangular matrix. This technique is used to simplify solving linear systems and is based on performing systematic elimination steps, where each step corresponds to multiplying by elementary lower triangular matrices.

Leading Principal Minors

The existence of an LU factorization without row exchanges is often contingent upon the condition that each leading principal minor of the matrix is nonzero. This is because the diagonal entries (pivots) in the factorization are determined by these minors, and a zero in any of them indicates a need for pivoting or that the matrix cannot be factorized in the standard form without modifications.

Pivoting

Pivoting refers to the strategy of row or column exchanges in order to overcome zero or insufficiently large pivot elements during factorization. If a matrix has a zero in a pivot position, standard LU factorization without pivoting fails, and one must either incorporate pivoting into the factorization process or conclude that a direct LU factorization in the standard form does not exist.

Transcript

00:01 Hello students, we are given a matrix here.

00:03 We need to show and we need to show that this matrix does not have an lu factorization, right? so what is lu factorization? l means lower triangular matrix and u means upper triangular matrix, right? so if i write l matrix, which is lower triangular matrix, so i can write this is lower.

00:23 So basically this is lower triangular matrix.

00:26 I can write this triangular matrix.

00:29 So in lower triangle matrix there is a condition and diagonal elements must be one and moreover the elements upper than the diagonal are zero and this element can be refer as this as this is the element of lower triangular matrix.

00:46 So i can write this element can be anything any value so this can be written as second row and first column element right? so if i define you as upper triangular matrix we can write this matrix as this is you can be written as so in upper triangle on matrix this value this lower value is always zero and these value can be anything u11 this is u22 and this is u12 and this is u12 right now a must be converted into l u factors so we can write l and u matrix in the form of u in l and u matrix we need to check in we need to solve it if it is not possible then there will be a condition which will we get after this right so this is 0 1 and this is 1 0 and l matrix can be 0 l 211 and this matrix is u11 and u1 2 2 right so here we need to calculate all the values and solve it so we need to multiply it and this is 0 1 and 10 matrix and this can be written as row must be multiplied to this column so this can be written as u and this is u12 right and if i talk about this term this element would be l21 u11 and again other element will be l 21 l21 l2 121 u1 2 plus u22 right so after comparison what we get if we want to compare two matrices then individual elements corresponding elements must be equal so u11 is 0 and u12 must be equal to 1 and if i what would be l21 multiplied by u1 is given as 1 right and according to other terms l21 u1 2 plus u2 m2 must be equals to again 0 right if i put a values here so we can find our answers right so u12 is already 1 so l21 plus u22 equals to 0 but here what we see here l2 1 multiplied by this is 0 so this is 0 equals to 1 and l21 equals to 1 divided by 0 so basically this stands to infinity which is undefined.

03:05 So we can easily predict that.

03:09 Hence, i can say, hence element of lower triangular matrix...

Please give Ace some feedback