Elementary Matrices and Row Operations

Overview

This lecture covers elementary matrices, their connection to elementary row operations, and methods for finding the inverse of matrices using both formulas and row operations. Examples are included for each concept.

Elementary Row Operations and Matrices

• Three elementary row operations:
  • Multiply a row by a nonzero constant (e.g., multiply Row 2 by 3).
  • Swap two rows (e.g., interchange Row 1 and Row 3).
  • Add a multiple of one row to another (e.g., add 2 × Row 1 to Row 2).
• Row equivalence:
  Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.
  Example:
  If A can be transformed into B by swapping rows and then scaling a row, A and B are row equivalent.
• Elementary matrix:
  An elementary matrix is obtained from the identity matrix by performing one elementary row operation.
  Example:
  Starting with the 2×2 identity matrix, if you multiply Row 2 by 5, you get
  [ \begin{bmatrix} 1 & 0 \ 0 & 5 \end{bmatrix} ] which is an elementary matrix.

Recognizing Elementary Matrices

• A matrix is elementary if it can be obtained from the identity matrix by a single elementary row operation.
  • Example:
    Swapping rows 1 and 2 of the 3×3 identity matrix gives
    [ \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} ]
• Matrices requiring more than one elementary operation are not elementary.
  • Example:
    If you need to both swap rows and scale a row to get from the identity to a matrix, it is not elementary.
• To return an elementary matrix to the identity, apply the inverse of the original row operation.
  • Example:
    If you multiplied Row 2 by 3 to get the elementary matrix, multiply Row 2 by 1/3 to return to the identity.

Using Elementary Matrices

• Multiplying an elementary matrix E by another matrix A (EA) applies the corresponding row operation to A.
  • Example:
    If E is the matrix that swaps rows 1 and 2, then EA is the result of swapping rows 1 and 2 of A.
• Multiplying is equivalent to performing the row operation directly on the matrix.
• The product of several elementary matrices represents a sequence of row operations.
  • Example:
    If E₁ scales a row and E₂ swaps rows, then E₂E₁A applies both operations to A in order.

Finding Inverses of Matrices

• The inverse of a 2×2 matrix exists only if the determinant is nonzero.
  Formula Example:
  For
  [ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ] the inverse is
  [ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} ] If ( ad-bc = 0 ), the matrix is not invertible.
• For larger matrices, augment with the identity and perform row operations to get the identity on the left; the right side then becomes the inverse.
  • Example:
    For a 3×3 matrix, write [A | I], use row operations to get [I | A⁻¹].

Invertibility and Elementary Matrices

• Every elementary matrix is invertible, and its inverse is also elementary.
  • Example:
    The inverse of a row swap is the same row swap; the inverse of multiplying a row by k is multiplying by 1/k.
• The process may be lengthy and error-prone for larger matrices; work carefully and use integer operations when possible to avoid fractions.
  • Example:
    When working with a 4×4 matrix, try to clear entries using integer multiples to keep calculations simpler.

Matrix Products as Elementary Matrices

• Any invertible matrix can be expressed as a product of elementary matrices.
  • Example:
    If you reduce a matrix to the identity using three row operations, then the original matrix equals the product of the inverses of the three corresponding elementary matrices.
• The steps to reduce a matrix to the identity via row operations correspond to multiplying by elementary matrices.

Key Terms & Definitions

• Elementary Row Operation: A basic manipulation: row scaling (e.g., multiply Row 1 by 2), swapping (e.g., switch Row 2 and Row 3), or adding multiples of rows (e.g., add 4 × Row 1 to Row 2).
• Row Equivalent: Two matrices are row equivalent if one can be transformed into the other via elementary row operations.
• Elementary Matrix: A matrix obtained from the identity by a single elementary row operation (e.g.,
  [ \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix} ] is obtained by adding 2 × Row 1 to Row 2 of the 2×2 identity).
• Invertible (Nonsingular) Matrix: A matrix with an inverse; determinant is nonzero.
• Determinant: A scalar value used to decide invertibility for square matrices (e.g., for 2×2, ( ad-bc )).

Action Items / Next Steps

• Complete the assigned homework problems, focusing on recognizing and applying elementary matrices and finding matrix inverses.
• Practice expressing matrices as products of elementary matrices.
• Review the equivalence of statements about invertibility from this section.
• Try creating your own examples of elementary matrices and their inverses for extra practice.