Matrix addition and scalar multiplication are basic operations for working with arrays of numbers. They matter because matrices are used to organize data, represent transformations, solve systems of equations, and model real situations in physics, engineering, computer graphics, and economics. These operations are simple when you track the position of each entry carefully.
The key idea is that each entry keeps its row and column address during the operation.
To add or subtract matrices, the matrices must have the same dimensions, such as both being 2 by 3 or both being 4 by 4. Matching entries are combined element by element, so the entry in row 1, column 2 only interacts with the entry in row 1, column 2 of the other matrix. In scalar multiplication, one number multiplies every entry in the matrix.
These rules let you build more complex matrix expressions while preserving the structure of the matrix.
Key Facts
- Matrices can be added only if they have the same dimensions.
- If A = [aij] and B = [bij], then A + B = [aij + bij].
- Matrix subtraction is also element by element: A - B = [aij - bij].
- Scalar multiplication multiplies every entry: kA = [k aij].
- The result of adding, subtracting, or scalar multiplying matrices has the same dimensions as the original matrices involved.
- Matrix addition is commutative and associative: A + B = B + A and (A + B) + C = A + (B + C).
Vocabulary
- Matrix
- A matrix is a rectangular array of numbers arranged in rows and columns.
- Entry
- An entry is one number inside a matrix, identified by its row and column position.
- Dimensions
- The dimensions of a matrix describe its size as number of rows by number of columns.
- Scalar
- A scalar is a single number that can multiply every entry of a matrix.
- Element-by-element operation
- An element-by-element operation combines entries that are in the same position in their matrices.
Common Mistakes to Avoid
- Adding matrices with different dimensions is wrong because there is no matching entry for every position.
- Multiplying only one row or one column by a scalar is wrong because scalar multiplication applies to every entry in the matrix.
- Combining entries that are not in the same position is wrong because matrix addition depends on matching row and column locations.
- Forgetting negative signs during subtraction is wrong because A - B means subtract each entry of B from the matching entry of A.
Practice Questions
- 1 Let A = [[2, -1], [4, 0]] and B = [[3, 5], [-2, 7]]. Find A + B.
- 2 Let C = [[1, -3, 2], [0, 4, -5]]. Find -2C.
- 3 Explain why the matrices [[1, 2, 3], [4, 5, 6]] and [[7, 8], [9, 10], [11, 12]] cannot be added, even though both contain 6 entries.