Section 2.5 Elementary Matrices
Definition: An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.
Example 1: [latex]E_{1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ -2 & 0 & 1 \end{bmatrix}[/latex], [latex]E_{2}=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}[/latex], [latex]E_{3}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{bmatrix}[/latex],
and [latex]A=\begin{bmatrix} x_{11} & x_{12} & x_{13}\\ x_{21} & x_{22} & x_{23}\\ x_{31} & x_{32} & x_{33} \end{bmatrix}[/latex]. [latex]E_{1}[/latex], [latex]E_{2}[/latex] and [latex]E_{3}[/latex] are elementary matrices.
Describe how to get [latex]E_{1}[/latex], [latex]E_{2}[/latex] and [latex]E_{3}[/latex] from identity matrix [latex]I_{3}[/latex] by elementary row operations. Compute [latex]E_{1}A[/latex], [latex]E_{2}A[/latex] and [latex]E_{3}A[/latex] and describe how these products can be obtained by elementary row operations.
Exercise 1: [latex]E_{1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & -3 & 1 \end{bmatrix}[/latex], [latex]E_{2}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix}[/latex], [latex]E_{3}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 3 \end{bmatrix}[/latex],
and [latex]A=\begin{bmatrix} x_{11} & x_{12} & x_{13}\\ x_{21} & x_{22} & x_{23}\\ x_{31} & x_{32} & x_{33} \end{bmatrix}[/latex]. [latex]E_{1}[/latex], [latex]E_{2}[/latex] and [latex]E_{3}[/latex] are elementary matrices.
Describe how to get [latex]E_{1}[/latex], [latex]E_{2}[/latex] and [latex]E_{3}[/latex] from identity matrix [latex]I_{3}[/latex] by elementary row operations. Compute [latex]E_{1}A[/latex], [latex]E_{2}A[/latex] and [latex]E_{3}A[/latex] and describe how these products can be obtained by elementary row operations.
Fact: 1. If an elementary row operation is performed on an [latex]m \times n[/latex] matrix [latex]A[/latex], the resulting matrix can be written as [latex]EA[/latex], where the [latex]m \times m[/latex] matrix [latex]E[/latex] is created by performing the same row operation on [latex]I_{m}[/latex].
2. Each elementary matrix is invertible. The inverse of [latex]E[/latex] is the elementary matrix of the same type that transforms [latex]E[/latex] back into [latex]I[/latex].
Theorem: An [latex]n \times n[/latex] matrix is invertible if and only if [latex]A[/latex] is row equivalent to [latex]I_{n}[/latex] and in this case, any sequence of elementary row operations that reduces [latex]A[/latex] to [latex]I_{n}[/latex] also transforms [latex]I_{n}[/latex] into [latex]A^{-1}[/latex].
Proof:
Fact: If [latex]E_{p}E_{p-1} \cdots E_{1}A = I_{n}[/latex] then [latex]A^{-1} = (E_{p} \cdots E_{1})I_{n}[/latex] where [latex]E_{i}'s[/latex] are elementary matrices that transform [latex]A[/latex] into [latex]I_{n}[/latex].
Example 2: Find [latex]A^{-1}[/latex] where [latex]A=\begin{bmatrix} 1 & 0 & 3\\ 0 & 1 & 1\\ 2 & -1 & 2 \end{bmatrix}[/latex].
Exercise 2: Find [latex]A^{-1}[/latex] where [latex]A=\begin{bmatrix