Tricks for remembering elementary matrix operations

May 29, 2024December 16, 2024by Kurious Fox

Connecting matrices to systems of linear equations can indeed help in better understanding and remembering the properties of matrices. By visualizing how matrix operations correspond to operations on systems of equations, abstract matrix properties become more concrete and intuitive. Here are a few key properties of matrices, illustrated with examples:

1. Matrix Addition and Subtraction

Property: Matrices of the same dimensions can be added or subtracted element-wise.

Example with Systems of Linear Equations:
Consider two systems of linear equations:

System 1:
$\begin{cases} 2x + 3y = 5 \\ 4x - y = 6 \end{cases}$

System 2:
$\begin{cases} 1x + 2y = 3 \\ 3x + y = 2 \end{cases}$

We can represent these systems with matrices (A) and (B):
$A = \begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix}$
$B = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix}$

Matrix Addition:
$A + B = \begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix} + \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 7 & 0 \end{pmatrix}$

Matrix Subtraction:
$A - B = \begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix} - \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & -2 \end{pmatrix}$

These operations correspond to adding or subtracting the corresponding coefficients in the systems of equations.

Elementary Matrix Operations and Their Corresponding Row Operations

Row Swapping (Elementary Matrix) Operation: Swap two rows of a matrix. Example: Consider the system of equations:
$\begin{cases} x + y = 2 \\ 2x + 3y = 5 \end{cases}$
The corresponding augmented matrix is:
$\begin{pmatrix} 1 & 1 & | & 2 \\ 2 & 3 & | & 5 \end{pmatrix}$
Swapping the two rows gives:
$\begin{pmatrix} 2 & 3 & | & 5 \\ 1 & 1 & | & 2 \end{pmatrix}$
This operation corresponds to using the elementary matrix:
$E = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
Multiplying this elementary matrix by the original matrix:
$E \begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 2 & 3 \\ 1 & 1 \end{pmatrix}$
Row Multiplication by a Scalar (Elementary Matrix) Operation: Multiply a row of the matrix by a non-zero scalar.
Example: Consider the system of equations:
$\begin{cases} x + y = 2 \\ 2x + 3y = 5 \end{cases}$
The corresponding augmented matrix is:
$\begin{pmatrix} 1 & 1 & | & 2 \\ 2 & 3 & | & 5 \end{pmatrix}$
Multiplying the first row by 3 gives:
$\begin{pmatrix} 3 & 3 & | & 6 \\ 2 & 3 & | & 5 \end{pmatrix}$
This operation corresponds to using the elementary matrix:
$E = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$
Multiplying this elementary matrix by the original matrix:
$E \begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 2 & 3 \end{pmatrix}$
Row Addition (Elementary Matrix) Operation: Add a multiple of one row to another row.
Example: Consider the system of equations:
$\begin{cases} x + y = 2 \\ 2x + 3y = 5 \end{cases}$
The corresponding augmented matrix is:
$\begin{pmatrix} 1 & 1 & | & 2 \\ 2 & 3 & | & 5 \end{pmatrix}$
Adding -2 times the first row to the second row gives:
$\begin{pmatrix} 1 & 1 & | & 2 \\ 0 & 1 & | & 1 \end{pmatrix}$
This operation corresponds to using the elementary matrix:
$E = \begin{pmatrix} 1 & 0 \\ -2 & 1 \end{pmatrix}$
Multiplying this elementary matrix by the original matrix:
$E \begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$