c11
Use row 1 of A, [1, 2, 0], and column 1 of B, [2, 0, 1].
(1 × 2) + (2 × 0) + (0 × 1) = 2
A (3×3)
B (3×3)
Enter matrices A and B, choose their dimensions, and calculate A × B. The number of columns in matrix A must equal the number of rows in matrix B. Supports integers, negative numbers, decimals, and fractions such as 1/2.
Calculation Results
3x3 Matrix Guide
3x3 multiplication follows the row-by-column rule. The product matrix has 9 entries, and each entry is one dot product between a row of Matrix A and a column of Matrix B.
Quick Explanation
A 3x3 matrix multiplication problem uses the row-by-column rule. To build the final product matrix, calculate 9 entries: each one comes from one row of A and one column of B.
When Two 3x3 Matrices Can Be Multiplied
Two 3x3 matrices can always be multiplied by each other because the inner dimensions match: the first matrix has 3 columns, and the second matrix has 3 rows.
The result is also a 3x3 matrix. That means you need to calculate 9 separate entries: c11, c12, c13, c21, c22, c23, c31, c32, and c33.
General Formula
cᵢⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + aᵢ₃b₃ⱼ
The index i tells you which row of A to use. The index j tells you which column of B to use. Multiply the three matching pairs, then add the products.
Existing Worked Example
We will keep the example matrices from this page and calculate every entry of AB.
We will calculate all nine entries step by step.
Matrix A
120
314
251
×
Matrix B
213
042
125
=
Result positions
c11c12c13
c21c22c23
c31c32c33
Calculate the First Row
c12
Use row 1 of A, [1, 2, 0], and column 2 of B, [1, 4, 2].
(1 × 1) + (2 × 4) + (0 × 2) = 9
c13
Use row 1 of A, [1, 2, 0], and column 3 of B, [3, 2, 5].
(1 × 3) + (2 × 2) + (0 × 5) = 7
Calculate the Second Row
c21
Use row 2 of A, [3, 1, 4], and column 1 of B, [2, 0, 1].
(3 × 2) + (1 × 0) + (4 × 1) = 10
c22
Use row 2 of A, [3, 1, 4], and column 2 of B, [1, 4, 2].
(3 × 1) + (1 × 4) + (4 × 2) = 15
c23
Use row 2 of A, [3, 1, 4], and column 3 of B, [3, 2, 5].
(3 × 3) + (1 × 2) + (4 × 5) = 31
Calculate the Third Row
c31
Use row 3 of A, [2, 5, 1], and column 1 of B, [2, 0, 1].
(2 × 2) + (5 × 0) + (1 × 1) = 5
c32
Use row 3 of A, [2, 5, 1], and column 2 of B, [1, 4, 2].
(2 × 1) + (5 × 4) + (1 × 2) = 24
c33
Use row 3 of A, [2, 5, 1], and column 3 of B, [3, 2, 5].
(2 × 3) + (5 × 2) + (1 × 5) = 21
Assemble the Result Matrix
Put each calculated entry into its matching position in the product matrix.
Final result matrix
AB
297
101531
52421
How Much Work Is Involved?
- Each result entry needs 3 scalar multiplications and 2 additions.
- A 3x3 product matrix has 9 result entries.
- Altogether, a full 3x3 multiplication uses 27 scalar multiplications and 18 additions.
Common 3x3 Mistakes
- Leaving out the third term in a dot product.
- Copying the wrong row from A or the wrong column from B.
- Putting a correct value into the wrong result position.
- Multiplying matrices entry by entry instead of row by column.
- Making a sign error in the middle of the arithmetic.
Calculator CTA
A full 3x3 multiplication has many small arithmetic steps. Use the main Matrix Multiplication Calculator when you want to check every entry, compare your work, or try larger matrices.
The calculator keeps the same row-by-column logic and gives you a faster way to verify the final product.
Related Guides
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.
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