Section 5.6 Elementary Matrices
Subsection 5.6.1 Elementary Matrices
Using the row perspective on matrix multiplication, we can characterize the elementary row operation as multiplication by a matrix.
Consider the case of a \(2 \times 2\) matrix,
\begin{equation*}
A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}
\end{equation*}
To multiply the top row by a scalar \(k\text{,}\) we can multiply by the matrix,
\begin{equation*}
\begin{bmatrix} k \amp 0 \\ 0 \amp 1 \end{bmatrix} \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} = \begin{bmatrix} ka \amp kb \\ c \amp d \end{bmatrix}
\end{equation*}
\(\begin{bmatrix} k \amp 0 \end{bmatrix}\)\(k\)\(A\text{,}\)\(0\)\(A\text{.}\)
\begin{equation*}
\begin{bmatrix} 1 \amp 0 \\ 0 \amp k \end{bmatrix} \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} = \begin{bmatrix} a \amp b \\ kc \amp kd \end{bmatrix}
\end{equation*}
To interchange the two rows, again consider the matrix,
\begin{equation*}
\begin{bmatrix} 0 \amp 1 \\ 1 \amp 0 \end{bmatrix} \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} = \begin{bmatrix} c \amp d \\ a \amp b \end{bmatrix}
\end{equation*}
Again, from the row perspective, the first row \(\begin{bmatrix} 0 \amp 1 \end{bmatrix}\) means taking 0 of the first row and 1 of the second row, to get \(\begin{bmatrix} c \amp d \end{bmatrix}\text{.}\)
For row replacement, that is, adding to one row a multiple of another row, again we use a linear combination. Suppose we want to add \(k R_2\) to \(R_1\text{.}\) In other words, \(R_1 + kR_2 \rightarrow R_1\text{.}\) Then,
\begin{equation*}
\begin{bmatrix} 1 \amp k \\ 0 \amp 1 \end{bmatrix} \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} = \begin{bmatrix} a + kc \amp b + kd \\ c \amp d \end{bmatrix}
\end{equation*}
These patterns and reasoning with row combinations generalize to arbitrary matrices. The idea is, the effect of a single row operation can be captured by left-multiplication by a matrix which is a slightly modified version of the identity matrix. In particular, the matrix that represents a particular elementary row operation is found by performing the same row operation on the identity matrix.
Theorem 5.6.2.
Let \(A\) be an \(m \times n\) matrix. If an elementary row operation is performed on \(A\text{,}\) the resulting matrix can be written as \(EA\) for an \(m \times n\) elementary matrix \(E\) formed by performing the same row operation on \(I_m\text{.}\)
In other words, every elementary row operation corresponds to left-multiplication by some elementary matrix.
In addition, every elementary matrix is invertible, intuitively because row operations are reversable. If \(E\) corresponds to some row operation, then the reverse operation has a corresponding elementary matrix \(F\) such that \(EF = FE = I\text{.}\)
In summary,
Theorem 5.6.3.
Every elementary matrix \(E\) is invertible, and the inverse of \(E\) is the elementary matrix of the same type as \(E\) which transforms \(E\) back into \(I\text{.}\)
Subsection 5.6.2 Permutation Matrices for Row Exchanges
The elementary matrices for row exchanges are the identity matrix with two row exchanged. For example, for \(2 \times 2\) matrices, there are 2 rows, so there is only 1 possible row exchange, represented by the matrix,
\begin{equation*}
\begin{bmatrix} 0 \amp 1 \\ 1 \amp 0 \end{bmatrix}
\end{equation*}
For \(3 \times 3\) matrices, there are 3 rows, and so \(3\) possible pairs of rows to exchange (\((1,2), (1,3)\text{,}\) and \((2,3)\)), represented by the matrices,
\begin{equation*}
\begin{array}{ccc}
(1,2) \amp (1,3) \amp (2,3) \\
\begin{bmatrix} 0 \amp 1 \amp 0 \\ 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \end{bmatrix} \amp \begin{bmatrix} 0 \amp 0 \amp 1 \\ 0 \amp 1 \amp 0 \\ 1 \amp 0 \amp 0 \end{bmatrix} \amp \begin{bmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \\ 0 \amp 1 \amp 0 \end{bmatrix}
\end{array}
\end{equation*}
In general, among \(n \times n\) matrices, there are \(n!\) permutation matrices.
