Matrix Operations: Grids of Numbers, Step by Step

A matrix is nothing more mysterious than a grid of numbers arranged in rows and columns. The ACT — unlike the SAT — includes a few matrix questions, and they're very learnable. Adding, subtracting, and scaling matrices are almost trivial once you see the pattern. Multiplication has exactly one rule worth memorizing, and the determinant of a 2×2 matrix is a one-line formula.

This guide walks through every matrix operation the ACT tests, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Matrices Matter (on the ACT)

Matrices appear on the ACT Math section but not on the SAT or SSAT, which makes them a distinctly ACT topic. The questions are usually straightforward computation, so a student who knows the rules can pick up quick, reliable points. They're also core Algebra II content.

What a Matrix Is

A matrix is a grid of numbers. Its dimensions are given as rows × columns: a matrix with 2 rows and 3 columns is a "2×3 matrix." Each number is an entry, located by its row and column.

Adding & Subtracting

To add or subtract two matrices, they must be the same size. Then just combine matching entries:

(Here [a b; c d] means a 2×2 matrix with top row a, b and bottom row c, d.) If the sizes differ, you can't add them.

Scalar Multiplication

Multiplying a matrix by a single number (a scalar) means multiplying every entry by that number:

Matrix Multiplication

This is the one with a rule. To multiply two matrices, take each row of the first times each column of the second, multiplying entry-by-entry and adding. For 2×2 matrices:

The top-left entry is (1·5 + 2·7) = 19; the top-right is (1·6 + 2·8) = 22, and so on.

The 2×2 Determinant

The determinant of a 2×2 matrix is a single number computed by cross-multiplying:

For [1 2; 3 4], the determinant is (1)(4) − (2)(3) = 4 − 6 = −2. The determinant shows up in solving systems and finding inverses.

Where You'll See This — Test by Test

Matrices are an ACT Math topic; the SAT and SSAT don't test them. No reference sheet provides the rules, so memorize the multiplication pattern and the determinant formula.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.