ACT® · Math: Number and Quantity
Matrix Addition, Subtraction, and Scalar Multiplication
Matrices add or subtract by combining entries in matching positions, so both matrices need identical dimensions. A scalar multiplies every entry by the same number.
A matrix is just a grid of numbers with a set number of rows and columns. Addition and subtraction work position by position: the entry in row 1, column 1 of the answer comes only from the row 1, column 1 entries of the two matrices you started with, and so on for every spot in the grid. That only works cleanly if every entry has a partner in the exact same spot in the other matrix. So the two matrices must have the same number of rows and the same number of columns. A 2 by 3 matrix and a 3 by 2 matrix both hold six numbers, but their grids are shaped differently, so entries don't line up position for position, and the sum is undefined even though the entry count matches.
Scalar multiplication is simpler: multiply every single entry by the same number. If a store's prices sit in a matrix and every price rises by 8 percent, you multiply the whole grid by 1.08, and each entry grows by that same factor. Combine the two ideas and you can compute something like 3A - B: first scale every entry of A by 3, then subtract B from the result entry by entry.
This sits right before matrix multiplication, which is a different animal entirely with its own rule about matching the number of columns in the first matrix to the number of rows in the second. Addition, subtraction, and scalar multiplication are the gentle warm-up: no new shape rules beyond matching dimensions, just careful bookkeeping across a grid.
Key terms
- Matrix
- A rectangular grid of numbers arranged in rows and columns.
- Entry
- A single number sitting in a specific row and column of a matrix.
- Scalar
- A single ordinary number that multiplies every entry of a matrix.
- Dimensions
- The number of rows and columns a matrix has, usually written rows by columns.
Watch it explained
- How it worksHow does adding two matrices run entry by entry, like stacking two spreadsheets?
- Worked exampleFind 3A minus B for A = [[2, -1], [0, 4]] and B = [[5, 3], [-2, 1]].
- Common mistakeWhy isn't a 2 by 3 matrix addable to a 3 by 2 matrix when both hold six numbers?
- Going furtherWhat happens to a matrix of store prices when the whole grid is multiplied by 1.08?
Check yourself
End of topic.