How To Find The Product Of Matrices: Step-by-Step Guide

How to Find the Product of Matrices

Ever stared at two grids of numbers and wondered how you're supposed to multiply them together? Day to day, here's the thing — once you see what you're actually doing, it clicks. Matrix multiplication is one of those topics that trips up a lot of people, not because it's impossibly hard, but because nobody explains it in plain language. You're not alone. And once it clicks, you can't unsee it.

So let's get into it.

What Is Matrix Multiplication?

Matrix multiplication is a way to combine two matrices (those rectangular arrays of numbers you see in brackets) to produce a new matrix. But here's what surprises most people: you're not multiplying each number in the first matrix by the corresponding number in the second. That would be element-wise multiplication, and it's not what matrix multiplication is.

Instead, you're taking rows from the first matrix and columns from the second matrix, multiplying them together in a specific way, and summing them up. This operation is sometimes called the dot product.

The result is a brand new matrix where each entry depends on an entire row from the first matrix and an entire column from the second one.

The Dimension Rule (This Is the Part Most People Miss)

Before you multiply anything, you need to check if multiplication is even possible. Here's the rule: the number of columns in the first matrix must equal the number of rows in the second matrix.

If Matrix A is 2×3 (2 rows, 3 columns) and Matrix B is 3×2 (3 rows, 2 columns), you can multiply them. Which means the inner dimensions (both 3) match. Your result will be a 2×2 matrix — the outer dimensions tell you the size of your answer.

But if you tried to multiply a 2×3 matrix by another 2×3 matrix? And that won't work. The inner dimensions (3 and 2) don't match, and you'll get an error or undefined result.

This check is step one. Skip it and you're just wasting time.

Why Matrix Multiplication Matters

You might be thinking, "Okay, cool math trick — but why should I care?"

Real talk: matrix multiplication is everywhere. Computer graphics use it to rotate, scale, and transform images. Machine learning algorithms run on matrix multiplication under the hood — neural networks are essentially massive chains of these operations. Physics, engineering, economics — anywhere there's structured data that needs to be combined in complex ways, matrices show up.

But even if you're just studying math, here's why it matters: matrix multiplication is how you solve systems of linear equations, transform geometric objects, and represent relationships in graph theory. It's a foundational skill. Master this, and a lot of other topics suddenly become easier.

How to Multiply Matrices (Step by Step)

Let's do this with actual numbers. Here's the process:

Step 1: Verify the Dimensions Work

Say we have Matrix A:

[1  2]
[3  4]

That's a 2×2 matrix (2 rows, 2 columns).

And Matrix B:

[5  6]
[7  8]

Also 2×2. Inner dimensions match (2 = 2), so we can multiply. The result will be 2×2.

Step 2: Find the Entry in Row 1, Column 1

Take the first row of Matrix A: [1, 2] Take the first column of Matrix B: [5, 7]

Now multiply corresponding entries and add them up: (1 × 5) + (2 × 7) = 5 + 14 = 19

So the top-left entry of your result is 19.

Here's what just happened: you took each number from the row, paired it with the matching-position number from the column, multiplied those pairs, and summed everything. That's the dot product.

Step 3: Find the Rest of the Entries

This pattern continues for every position:

• Row 1, Column 2: First row of A [1, 2] × Second column of B [6, 8] = (1 × 6) + (2 × 8) = 6 + 16 = 22

• Row 2, Column 1: Second row of A [3, 4] × First column of B [5, 7] = (3 × 5) + (4 × 7) = 15 + 28 = 43

• Row 2, Column 2: Second row of A [3, 4] × Second column of B [6, 8] = (3 × 6) + (4 × 8) = 18 + 32 = 50

Your final result:

[19  22]
[43  50]

That's it. You've multiplied two matrices.

A Larger Example (Just to Prove It Scales)

Let's try matrices with different dimensions to show the process works the same way.

Matrix A (2×3):

[1  2  3]
[4  5  6]

Matrix B (3×2):

[7  8]
[9  1]
[2  3]

We can multiply because the inner dimensions match (3 = 3). The result will be 2×2.

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Entry (1,1): Row 1 of A [1, 2, 3] × Column 1 of B [7, 9, 2] = (1×7) + (2×9) + (3×2) = 7 + 18 + 6 = 31

Entry (1,2): Row 1 of A [1, 2, 3] × Column 2 of B [8, 1, 3] = (1×8) + (2×1) + (3×3) = 8 + 2 + 9 = 19

Entry (2,1): Row 2 of A [4, 5, 6] × Column 1 of B [7, 9, 2] = (4×7) + (5×9) + (6×2) = 28 + 45 + 12 = 85

Entry (2,2): Row 2 of A [4, 5, 6] × Column 2 of B [8, 1, 3] = (4×8) + (5×1) + (6×3) = 32 + 5 + 18 = 55

Result:

[31  19]
[85  55]

Same process. Bigger numbers. No extra complexity.

Common Mistakes (And How to Avoid Them)

The most frequent errors I've seen:

Forgetting to check dimensions. I already said it, but it bears repeating: always verify the inner dimensions match before you start. This one mistake causes more frustration than anything else.

Multiplying in the wrong order. Matrix multiplication is not commutative — AB doesn't equal BA in general. Sometimes BA isn't even defined if the dimensions don't work in that direction. Be careful about which matrix comes first.

Adding instead of multiplying within each pair. When you compute a dot product, you're multiplying pairs of numbers, then adding the results. Students sometimes multiply the whole row by the whole column and get confused. Stick to: multiply corresponding entries, then sum.

Keeping track of which row and column go together. It helps to say it out loud or write it down: "I'm using row i from the first matrix and column j from the second matrix." Otherwise it's easy to lose your place.

Practical Tips That Actually Help

• Use paper, not just your head. Writing out the rows and columns side by side prevents mistakes. Some people even draw arrows to connect the row to the column they're working with.

• Color-code if that helps. Highlighting rows in one color and columns in another makes it visually obvious what you're pairing.

• Check your work by estimating. If you multiplied two positive matrices and got a negative number in the middle of your result, something probably went wrong. Matrix multiplication with all positive entries yields all positive entries (for the standard definition).

• Practice with small matrices first. Master 2×2 multiplication until it's automatic. Then 2×3 by 3×2. The process is identical — you just have more entries to compute.

• Remember: it's row times column. Someone asks you to describe matrix multiplication in one phrase, that's it. Row times column.

FAQ

Can you multiply any two matrices?

No. Still, the number of columns in the first matrix must equal the number of rows in the second matrix. If that condition isn't met, the multiplication is undefined.

Does matrix multiplication commute? That is, does AB = BA?

Generally, no. Matrix multiplication is not commutative. In special cases (like when both matrices are identity matrices or scalar multiples of the identity), they might commute, but you can't assume this. Also, even when both products exist, they're usually different.

What's the identity matrix?

It's a square matrix with 1s on the diagonal and 0s everywhere else. Multiplying any matrix by the identity (of the right size) gives you back the original matrix. Think of it like multiplying by 1 in regular arithmetic.

How do I multiply a matrix by a scalar?

That's different from matrix multiplication. A scalar is just a regular number. To multiply a matrix by a scalar, you multiply every single entry in the matrix by that number. It's much simpler — no rows and columns to worry about.

Can I use a calculator for matrix multiplication?

Yes — graphing calculators (like TI-84s) have matrix functions, and apps like Desmos can handle it. But if you're learning, do it by hand first. The practice builds intuition, and you'll understand what's actually happening when the calculator gives you an answer.

The Short Version

Matrix multiplication is a systematic process: check your dimensions, then for each entry in your result, take the corresponding row from the first matrix and the corresponding column from the second, multiply matching pairs, and add them up. That's it.

The reason this topic seems harder than it is comes down to two things: people skip the dimension check, and they lose track of which row pairs with which column. Fix those two issues and suddenly it's straightforward.

So grab a piece of paper, pick two matrices, and work through it. The first one takes effort. The second one starts to feel natural. By the third, you'll wonder what the fuss was about.