# What is elementary matrix example?

Table of Contents

## What is elementary matrix example?

The matrix M is called a left-inverse of A because when it is multiplied to the left of A, we get the identity matrix….Introducing the left inverse of a square matrix.

Matrix | Elementary row operation | Elementary matrix |
---|---|---|

[102−1010−1001−1] | R1←R1+(−2)R3 | M4=[10−2010001] |

[1001010−1001−1] |

### How do you know if a matrix has a LU factorization?

A square matrix is said to have an LU decomposition (or LU factorization) if it can be written as the product of a lower triangular (L) and an upper triangular (U) matrix.

#### Is LU factorization always possible?

LUP always exists (We can use this to quickly figure out the determinant). If the matrix is invertible (the determinant is not 0), then a pure LU decomposition exists only if the leading principal minors are not 0.

**How do you find L and U in LU decomposition?**

LU Decomposition Method or Factorisation

- Step 1: Generate a matrix A = LU such that L is the lower triangular matrix with principal diagonal elements being equal to 1 and U is the upper triangular matrix.
- Step 2: Now, we can write AX = B as:
- Step 3: Let us assume UX = Y….(2)
- Step 4: From equations (1) and (2), we have;

**Why do we use LU factorization?**

LU decomposition is a better way to implement Gauss elimination, especially for repeated solving a number of equations with the same left-hand side. That is, for solving the equation Ax = b with different values of b for the same A.

## What is LU factorization in solving a system of linear equations?

LU decomposition of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix.

### Which is an elementary matrix?

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field.

#### What is the determinant of elementary matrix?

The determinant of an elementary matrix E is given as follows: (a). det (E) = -1, if E interchanges two rows. (b). det (E) = C, if E multiplies a row by a non-zero constant c.

**Which is the elementary matrix?**

**Can you Lu Factorize a non square matrix?**

For matrices that are not square, LU decomposition still makes sense. Given an m × n matrix M, for example we could write M = LU with L a square lower unit triangular matrix, and U a rectangular matrix. Then L will be an m × m matrix, and U will be an m × n matrix (of the same shape as M).