First result entry: c11
Take the first row of A, [1, 2, 0], and the first column of B, [2, 0, 1].
(1 x 2) + (2 x 0) + (0 x 1) = 2
3x3 Matrix Guide
3x3 Matrix Multiplication Step by Step
3x3 multiplication follows the same row-by-column rule as 2x2 multiplication, but each answer entry takes more arithmetic. Once you keep the rows and columns organized, the process becomes much easier to follow.
Why 3x3 Matrix Multiplication Is More Complex
A 3x3 problem uses the same rule as smaller matrices, but each answer entry needs three multiplications and two additions.
There are also more entries in the final matrix, so accuracy matters more because a small row-or-column mistake can affect the whole result.
3x3 Matrix Multiplication Formula
Cij = ai1b1j + ai2b2j + ai3b3j
In simple terms, pick one row from A and one column from B, multiply the matching entries, and add all three products.
How to Multiply a 3x3 Matrix Step by Step
This 3x3 matrix multiplication example shows the full matrices first so you can see the complete row-by-column structure before calculating the result.
We’ll compute the first two entries step by step.
Matrix A
120
314
251
x
Matrix B
213
042
125
Second result entry: c12
Keep the same first row of A, then use the second column of B, [1, 4, 2].
(1 x 1) + (2 x 4) + (0 x 2) = 9
The same row-by-column process is repeated for the remaining entries.
The remaining entries are found the same way: choose the correct row from A, match it
with the correct column from B, multiply the three pairs, and add the products.
Final result matrix
AB
297
101531
52421
Tip for Solving 3x3 Matrices
Work in a fixed order, such as across the first row of the result matrix before moving down to the next row. That makes it much easier to keep the correct row and column paired together.
Use the Matrix Multiplication Calculator
If you want quick answers, want to check your arithmetic, or want to multiply larger matrices, the main calculator is the easiest next step.
3x3 calculations can get long quickly — use the calculator to save time.
Related Matrix Multiplication Pages
Frequently Asked Questions
How do you multiply 3x3 matrices?
Multiply each row of the first 3x3 matrix by each column of the second 3x3 matrix. Each result entry uses three multiplications and then adds those products together.
Why is 3x3 matrix multiplication harder than 2x2?
It is harder because each entry has more terms and there are more entries to compute, so it is easier to lose track of rows, columns, or additions.
Can I use a calculator for 3x3 matrices?
Yes. A matrix multiplication calculator can quickly solve 3x3 problems, help you check your work, and make larger matrix multiplication easier.
Learn matrix multiplication more easily in just 2 minutes with this free game.
Solve your matrix instantly with our free calculator.
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