Matrix questions on the ACT are surprisingly formulaic. Once you learn the three operations, you can solve every matrix question in under 60 seconds.
What Is a Matrix?
A matrix is a rectangular grid of numbers. A matrix has 2 rows and 3 columns:
Operation 1: Addition and Subtraction
Add or subtract corresponding entries. The matrices must be the same size.
Operation 2: Scalar Multiplication
Multiply every entry by the scalar:
Operation 3: Matrix Multiplication
This is the only tricky one. For :
Rule: Number of columns of must equal number of rows of .
Result size: If is and is , result is .
How to compute each entry: Row of times column of (dot product).
The Dimension Check
Before multiplying, always verify: inner dimensions must match.
→ inner dimensions are both 3 ✓ → result is .
→ inner dimensions are 3 and 2 ✗ → multiplication impossible.
Common Mistakes to Avoid
- : Matrix multiplication is NOT commutative. Order matters.
- Adding different-sized matrices: plus is undefined.
- Dot product errors: Row times column means multiply corresponding entries and ADD the products.
ACT Pro Tip
The ACT almost always gives you small matrices ( or ). There are at most 4 entries to compute. Work methodically: row 1 times column 1, row 1 times column 2, row 2 times column 1, row 2 times column 2. Four quick calculations and you are done.
Practice matrices with our Matrices lesson — ACT-exclusive content.