Elementary matrices

Let $ I_n$ be the $ n\times n$ identity matrix. The matrix $ E$ obtained by either

  1. adding a multiple of the $ i$-th row of $ I_n$ by $ k$ to the $ j$-th row of $ I_n$, or
  2. interchanging the $ i$-th row and the $ j$-th row of $ I_n$, or
  3. multiplying the $ i$-th row of $ I_n$ by $ c$,
is called an elementary matrix. Given an $ n\times m$ matrix $ A$, the matrix multiplication $ E A$ results in performing the corresponding basic row operation on $ A$. Furthermore, the inverse matrix $ E^{-1}$ of the elementary matrix $ E$ is also the elementary matrix obtained respectively by
  1. adding a multiple of the $ i$-th row of $ I_n$ by $ (-k)$ to the $ j$-th row of $ I_n$,
  2. interchanging the $ i$-th row and the $ j$-th row of $ I_n$ (thus, the same matrix),
  3. multiplying the $ i$-th row of $ I_n$ by $ c^{-1}$.

EXAMPLE 7. Find the $ 3\times 3$ elementary matrix corresponding to each of the following row operations:

  1. Add a multiple of the 1st row by -4 to the 3rd row.
  2. Interchange the 1st and 2nd row.
  3. multiplying the 3rd row by 5.

EXAMPLE 8. Find the inverse of $ E_1 =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-4 & 0 & 1
\end{bmatrix}$

Finding $ A^{-1}$. Let $ A$ be a square matrix. Suppose that $ A \sim I_n$. Then we have a series of basic row operations which reduces $ A$ to $ I_n$. Or, equivalently we have a series of elementary matrices  $ E_1, E_2, \ldots, E_p$ so that $ E_p \cdots E_2 E_1 A = I_n$. Comparing

$\displaystyle (E_p \cdots E_2 E_1) A = I_n$    and $\displaystyle \quad
A^{-1} A = I_n,
$

we have $ A^{-1} = E_p \cdots E_2 E_1$.



Department of Mathematics
Last modified: 2005-10-21