1. What Is Matrix Subtraction?
Matrix subtraction means subtracting one matrix from another by subtracting their corresponding elements. Just like addition works entry-by-entry, subtraction also works position-by-position.
If you can subtract numbers in a table, you can subtract matrices because the idea is exactly the same.
2. Condition for Subtracting Two Matrices
You can subtract two matrices only if they have the same order. This means both matrices must have:
- the same number of rows
- the same number of columns
If their sizes do not match, subtraction is not possible.
\( A - B \text{ is defined only if } A \text{ and } B \text{ are both } m \times n. \)
3. Rule for Matrix Subtraction
If \( A = [a_{ij}] \) and \( B = [b_{ij}] \) are two matrices of the same order, then their difference \( A - B \) is the matrix obtained by subtracting corresponding entries:
\( (A - B)_{ij} = a_{ij} - b_{ij} \)
This rule is applied to every element in the matrix.
4. Example: Subtracting Two Matrices
Let:
\( A = \begin{bmatrix} 6 & 2 \\ 4 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix} \)
Both matrices are \( 2 \times 2 \), so subtraction is allowed.
4.1. Subtracting Corresponding Elements
- \( 6 - 3 = 3 \)
- \( 2 - (-1) = 3 \)
- \( 4 - 5 = -1 \)
- \( -1 - 2 = -3 \)
4.2. Final Result
\( A - B = \begin{bmatrix} 3 & 3 \\ -1 & -3 \end{bmatrix} \)
5. Another Example
Subtract the matrices:
\( X = \begin{bmatrix} 4 & -2 & 5 \\ 7 & 3 & 1 \end{bmatrix}, \quad Y = \begin{bmatrix} 1 & 0 & 2 \\ 4 & 8 & -3 \end{bmatrix} \)
5.1. Corresponding Differences
Subtract entry by entry:
- \( 4 - 1 = 3 \)
- \( -2 - 0 = -2 \)
- \( 5 - 2 = 3 \)
- \( 7 - 4 = 3 \)
- \( 3 - 8 = -5 \)
- \( 1 - (-3) = 4 \)
5.2. Final Matrix
\( X - Y = \begin{bmatrix} 3 & -2 & 3 \\ 3 & -5 & 4 \end{bmatrix} \)
6. Observations About Matrix Subtraction
Matrix subtraction follows familiar arithmetic patterns:
- Not commutative: \( A - B \neq B - A \) in general.
- Compatible with addition: \( A - B = A + (-B) \)
- Zero matrix acts as an identity: \( A - O = A \)
These properties make it easier to handle subtraction when solving matrix equations or simplifying expressions.