Matrix Multiplication Calculator (3×3)
Enter matrices A and B row by row.
Your numbers
Your results
A × B, row 1
A × B, row 1
—
A × B, row 2
—
A × B, row 3
—
det(A × B)
—
Trace of A × B
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Matrix Multiplication Calculator (3×3) by TheCalcTools
How the Matrix Multiplication Calculator (3×3) works
Each entry of the product is the dot product of a row of A with a column of B: (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ. Matrix multiplication is not commutative: A × B usually differs from B × A.
The determinant of a product is the product of the determinants, a useful check.
The formula
(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ; det(AB) = det(A) det(B)
Worked example
Using the default values shown in the calculator:
| A row 1, column 1 | 2 |
|---|---|
| A row 1, column 2 | -1 |
| A row 1, column 3 | 0 |
| A row 2, column 1 | 1 |
| A row 2, column 2 | 3 |
| A row 2, column 3 | 2 |
| A row 3, column 1 | 0 |
| A row 3, column 2 | 1 |
| A row 3, column 3 | 4 |
| B row 1, column 1 | 1 |
| B row 1, column 2 | 0 |
| B row 1, column 3 | 2 |
| B row 2, column 1 | 3 |
| B row 2, column 2 | -1 |
| B row 2, column 3 | 1 |
| B row 3, column 1 | 0 |
| B row 3, column 2 | 2 |
| B row 3, column 3 | 5 |
- Row 1 of A × B: [ -1, 1, 3 ].
- det(A × B) = 120.0000.
Result:
| A × B, row 1 | [ -1, 1, 3 ] |
|---|---|
| A × B, row 2 | [ 10, 1, 15 ] |
| A × B, row 3 | [ 3, 7, 21 ] |
| det(A × B) | 120.0000 |
| Trace of A × B | 21.0000 |
Frequently asked questions
Can I multiply a 2×3 by a 3×2 matrix?
Yes in general, but this calculator handles square 3×3 (or 2×2 padded with zeros) matrices.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.