Matrix Subtraction
Perform matrix subtraction in Grade 10 algebra. Subtract two same-dimension matrices by adding the additive inverse of each element, computing entry-by-entry differences.
Key Concepts
Property To subtract two matrices of the same dimensions, $A B$, take the opposite, or additive inverse, of $B$ and add it to $A$. $$ \begin{bmatrix} a {11} & a {12} \\ a {21} & a {22} \end{bmatrix} \begin{bmatrix} b {11} & b {12} \\ b {21} & b {22} \end{bmatrix} = \begin{bmatrix} a {11} + ( b {11}) & a {12} + ( b {12}) \\ a {21} + ( b {21}) & a {22} + ( b {22}) \end{bmatrix} $$.
$$ \begin{bmatrix} 5 & 10 \\ 20 & 0 \end{bmatrix} \begin{bmatrix} 2 & 8 \\ 15 & 12 \end{bmatrix} = \begin{bmatrix} 5 & 10 \\ 20 & 0 \end{bmatrix} + \begin{bmatrix} 2 & 8 \\ 15 & 12 \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ 5 & 12 \end{bmatrix} $$ $$ \text{To solve } X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 10 & 10 \\ 10 & 10 \end{bmatrix}, \text{ find } X = \begin{bmatrix} 10 & 10 \\ 10 & 10 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 9 & 8 \\ 7 & 6 \end{bmatrix} $$.
Matrix subtraction is a clever trick! Instead of actually subtracting, you flip the signs of every number in the second matrix to find its 'additive inverse' and then simply add them like usual. This method transforms a subtraction problem into an addition one, which makes calculations much simpler and helps you avoid those pesky sign errors.
Common Questions
How do you subtract two matrices?
Subtract A - B by adding the opposite of each element in B to the corresponding element in A. Each entry in the result equals the corresponding entry of A minus that of B.
What requirement must matrices meet for subtraction?
Matrices must have the same dimensions (same number of rows and columns). You cannot subtract a 2×3 matrix from a 2×2 matrix.
How is matrix subtraction used in real-world applications?
Matrix subtraction is used to find differences between datasets, such as comparing budget vs. actual spending across departments, or tracking changes in data between two time periods.