Adding and Subtracting Matrices Revision Notes for VCE SSCE General Mathematics

Adding and Subtracting Matrices

Introduction to matrix addition and subtraction

When working with matrices, we can combine them using addition and subtraction operations. However, these operations follow specific rules that are different from simply adding or subtracting numbers. The key requirement is that the matrices must have the same order, meaning they must have the same number of rows and columns.

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Two matrices have the "same order" when they have identical dimensions - both the number of rows and the number of columns must match. For example, a $2 \times 3$ matrix can only be added to or subtracted from another $2 \times 3$ matrix.

Rules for adding and subtracting matrices

Understanding how to add and subtract matrices correctly is essential for working with these mathematical objects. Here are the three fundamental rules:

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Rule 1: Adding corresponding elements

When adding matrices, add each element to the element in the corresponding position. For example, the element in row 1, column 1 of the first matrix is added to the element in row 1, column 1 of the second matrix.

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Rule 2: Subtracting corresponding elements

When subtracting matrices, subtract each element from the element in the corresponding position. The element in each position of the second matrix is subtracted from the element in the same position of the first matrix.

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Rule 3: Same order requirement

Matrix addition and subtraction is only possible when both matrices have the same order. This means they must have identical dimensions - the same number of rows and the same number of columns. If the matrices have different orders, addition or subtraction cannot be performed.

Worked example: Adding matrices

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Worked Example: Adding $2 \times 2$ Matrices

Let's work through an example of adding two $2 \times 2$ matrices:

$\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} + \begin{bmatrix} 9 & 8 \\ 9 & -1 \end{bmatrix}$

Step 1: Write out the addition

$\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} + \begin{bmatrix} 9 & 8 \\ 9 & -1 \end{bmatrix}$

Step 2: Add the elements in the same positions

$= \begin{bmatrix} 2 + 9 & 4 + 8 \\ 5 + 9 & 1 + (-1) \end{bmatrix}$

Step 3: Evaluate each element

$= \begin{bmatrix} 11 & 12 \\ 14 & 0 \end{bmatrix}$

Notice how each element in the result comes from adding the corresponding elements in the original matrices. The top-left element is $2 + 9 = 11$ , the top-right is $4 + 8 = 12$ , and so on.

Worked example: Subtracting matrices

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Worked Example: Subtracting $3 \times 2$ Matrices

Now let's look at an example of subtracting two $3 \times 2$ matrices:

$\begin{bmatrix} 7 & 3 \\ 2 & 8 \\ 1 & 0 \end{bmatrix} - \begin{bmatrix} 4 & 2 \\ -1 & 9 \\ 3 & 7 \end{bmatrix}$

Step 1: Write out the subtraction

$\begin{bmatrix} 7 & 3 \\ 2 & 8 \\ 1 & 0 \end{bmatrix} - \begin{bmatrix} 4 & 2 \\ -1 & 9 \\ 3 & 7 \end{bmatrix}$

Step 2: Subtract the elements in the same positions

$= \begin{bmatrix} 7 - 4 & 3 - 2 \\ 2 - (-1) & 8 - 9 \\ 1 - 3 & 0 - 7 \end{bmatrix}$

Step 3: Evaluate each element

$= \begin{bmatrix} 3 & 1 \\ 3 & -1 \\ -2 & -7 \end{bmatrix}$

Pay special attention to the subtraction involving negative numbers. In the middle row, left column, we have $2 - (-1) = 2 + 1 = 3$ .

The zero matrix

A zero matrix is a special type of matrix where every element equals zero. This type of matrix is also called a null matrix. Zero matrices can have any order, and they play a similar role in matrix arithmetic as the number zero does in regular arithmetic.

Here are some examples of zero matrices with different orders:

$1 \times 1$ zero matrix: $[0]$
$2 \times 2$ zero matrix: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
$1 \times 2$ zero matrix: $[0 \quad 0]$
$2 \times 1$ zero matrix: $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$
$2 \times 3$ zero matrix: $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

Properties of the zero matrix

The zero matrix behaves similarly to the number zero in ordinary arithmetic. Understanding these properties helps solidify your understanding of matrix operations:

Adding a zero matrix: When you add a zero matrix to any matrix, the result is the original matrix unchanged.

$\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$

Subtracting a zero matrix: When you subtract a zero matrix from any matrix, the result is the original matrix unchanged.

$\begin{bmatrix} 5 \\ 6 \end{bmatrix} - \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}$

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Subtracting a matrix from itself

When you subtract any matrix from itself, the result is always a zero matrix. For example:

$[9 \quad 4 \quad 8] - [9 \quad 4 \quad 8] = [0 \quad 0 \quad 0]$

This makes sense because subtracting any number from itself gives zero, so subtracting each element from itself gives zero in every position.

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Key Points to Remember:

Same order required: You can only add or subtract matrices if they have the same order (same dimensions).
Element-by-element operations: Add or subtract corresponding elements in the same positions to create the result matrix.
Zero matrix definition: A zero matrix contains only zeros and acts like the number zero in regular arithmetic.
Zero matrix properties: Adding or subtracting a zero matrix leaves the original matrix unchanged, and subtracting any matrix from itself produces a zero matrix.
Watch your signs: Be careful with negative numbers when subtracting matrices - remember that subtracting a negative is the same as adding a positive.

Adding and Subtracting Matrices (VCE SSCE General Mathematics): Revision Notes