Adding two 2×2 matrices
[[1, 2], [3, 4]] + [[5, 6], [7, 8]] = [[6, 8], [10, 12]].
Matrix calculator
Result
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How it works
Addition and subtraction combine matrices of the same shape entry by entry. Multiplication is different: each entry of the result is a sum of products between a row of A and a column of B, which is why the inner dimensions have to match (A's columns = B's rows) even when the outer shape does not.
The determinant and inverse only make sense for a square matrix. The determinant is computed by cofactor expansion, which generalizes the familiar 2×2 shortcut (ad − bc) to any size. The inverse exists only when the determinant is nonzero — a zero determinant means the matrix collapses information (its rows or columns are not truly independent), which cannot be undone.
Formulas used
2×2 determinant
det = ad − bc, for matrix [[a, b], [c, d]]
Inverse
A⁻¹ = adj(A) ÷ det(A)
Worked examples
The determinant of a 2×2 matrix
The determinant of [[1, 2], [3, 4]] is 1×4 − 2×3 = −2.
Assumptions and limits
- Matrices are limited to 8×8 to keep the determinant and inverse calculations fast.
- All entries must be plain numbers — no fractions or expressions inside the matrix text.
Frequently asked questions
How do I format the matrix input?
One row per line, with the values in that row separated by commas or spaces — for example, a 2×3 matrix is two lines, each with three numbers.
Why does matrix multiplication need matching dimensions?
Each result entry sums products of a full row of A against a full column of B, so A's row length must match B's column length — if they don't line up, the sum simply cannot be formed.
What does a determinant of zero mean?
It means the matrix is singular: its rows (or columns) are linearly dependent, so it cannot be inverted, and the linear system it represents does not have a single unique solution.
Is matrix multiplication the same as multiplying entry by entry?
No — that operation exists (called the Hadamard product) but is different and much less common; standard matrix multiplication is the row-by-column sum-of-products described above, and this calculator uses that standard definition.
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