Elementary Matrices
Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations
◇Overview
Definition: An n×n matrix is called an elementary matrix if it can be obtained from the identity matrix In by performing a single Elementary Row O...
◇Key Concepts
Definition Invertibility Theorem Theorem (Equivalent Statements)
◇Details / Theory
Definition: An n×n matrix is called an elementary matrix if it can be obtained from the identity matrix In by performing a single ◇Elementary Row Operation. (Handout 2, Page 16)
sw operation is performed on an m×n matrix A, the resulting matrix can be obtained by premultiplying A by the corresponding m×m elementary matrix. (Handout 2, Page 17)
EA = result of applying the same row operation (that formed E from I) to A.
Invertibility: Elementary matrices are always nonsingular (invertible), and their inverses are also elementary matrices. (Handout 2, Page 18)
Inverse of Ri↔Rj is Ri↔Rj.
Inverse of kRi→Ri is k1Ri→Ri.
Inverse of Ri+kRj→Ri is Ri−kRj→Ri. Theorem: A square matrix A is nonsingular if and only if it can be expressed as a product of elementary matrices. (Handout 2, Page 18) Theorem (Equivalent Statements): For an n×n matrix A, the following are equivalent: (Handout 2, Page 19)
- A is nonsingular (invertible).
- The homogeneous system Ax=0 has only the trivial solution (x=0).
- A is row equivalent to In. (i.e., A can be reduced to In by elementary row operations).
- A can be written as a product of elementary matrices.
- The system Ax=b has a unique solution for every n×1 matrix b.
- The determinant of A is non-zero (det(A)=0). (This is a key result, though determinants might be covered later).
◇Common Pitfalls
Each elementary matrix corresponds to exactly one ERO — don’t combine multiple operations into one matrix. Elementary matrices are always square and always invertible.
◇Summary
TL;DR — An elementary matrix is obtained by performing one ERO on the identity. Left-multiplying by applies that same ERO. Elementary matrices are always invertible.
◇Links
Related: - ◇Elementary Row Operation Source / Reference:
◇References & Navigation
- Previous Topic: ◇Diagonal Matrix
- Next Topic: ◇Elementary Row Operation
- Related Notes: ◇Zero Matrix, ◇Matrix Addition, ◇Matrices Detailed
ACTIVE RECALL VERIFICATION
Select an answer to verify conceptual comprehension and reveal the technical rationale.
How is an elementary matrix constructed?
Which of the following statements about the inverse of an elementary matrix is true?
According to the equivalent statements for an matrix , if is nonsingular, which of the following must also be true?