Elementary Matrices

Chapter 2 • Section 2-5

"The building blocks of row reduction! 🧱"

🧱 What is an Elementary Matrix?

An elementary matrix is obtained by performing a single elementary row operation on the Identity Matrix ($I$).

Just one operation! No more, no less.

Interchange two rows.

$$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$

Multiply a row by $k \neq 0$.

$$ \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} $$

Add multiple of one row to another.

$$ \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} $$

Is $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ an elementary matrix?

Hint: Try to get it from $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ in one step.

If $E$ is an elementary matrix, then multiplying $EA$ performs the same row operation on $A$.

Example:

Let $E = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ (Swap rows).

Then $EA$ will swap the rows of $A$!

Matrix Multiplication

Left-multiplying acts on Rows.

(Right-multiplying acts on Columns!)

Every elementary row operation is reversible. Therefore:

And its inverse is an elementary matrix of the same type.

Swap Inverse:

Swap back! ($E^{-1} = E$)

Scale Inverse:

Multiply by $1/k$.

Add Inverse:

Subtract the multiple.

An $n \times n$ matrix $A$ is invertible if and only if $A$ can be written as a product of elementary matrices.

Let $A = \begin{bmatrix} 0 & 1 \\ 1 & 3 \end{bmatrix}$. Reduce to $I$:

Swap $R_1, R_2$: $E_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \implies E_1 A = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$.
$R_1 \leftarrow R_1 - 3R_2$: $E_2 = \begin{bmatrix} 1 & -3 \\ 0 & 1 \end{bmatrix} \implies E_2 (E_1 A) = I$.

So $E_2 E_1 A = I$. Thus $A = E_1^{-1} E_2^{-1}$.

$$ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} $$

If $A$ is invertible, can we write $A$ as a product of elementary matrices?