Choose a matrix size and an operation to add, subtract, multiply, transpose, or find the determinant of a matrix.
How to Use the Matrix Calculator
Pick a matrix size — 2×2 or 3×3 — and an operation first; that combination decides which grids show up. Addition, subtraction, and multiplication need two matrices, entered as Matrix A and Matrix B. Determinant and transpose only touch Matrix A. Type values into each cell and the result updates as you go.
Addition and subtraction work exactly like you'd guess: matching positions add or subtract, nothing more involved.
[1 2] [5 6] [ 6 8]
[3 4] + [7 8] = [10 12]Multiplication doesn't follow that pattern, and it trips up almost everyone meeting matrices for the first time. It is not componentwise. The entry in row i, column j of the product comes from the dot product of row i in Matrix A and column j in Matrix B — multiply corresponding entries, then add them:
[1 2] [5 6] [1×5+2×7 1×6+2×8] [19 22]
[3 4] × [7 8] = [3×5+4×7 3×6+4×8] = [43 50]Both matrices need to be the same N×N size for this calculator to multiply them — mismatched dimensions are a separate can of worms this tool doesn't try to open.
What a Determinant Actually Tells You
For a 2×2 matrix, the determinant is ad − bc. For 3×3, it's cofactor expansion along the first row: each entry in that row gets multiplied by the determinant of the 2×2 matrix left over when you cross out its row and column, alternating sign as you move across.
It isn't just a number that falls out of a formula. A 2×2 matrix's determinant is the factor by which the linear transformation it represents scales area — an identity matrix's determinant of 1 means area in equals area out, and a negative determinant flips orientation while still scaling by its absolute value. A determinant of zero means the transformation flattens the plane (or, for 3×3, space) down into a lower dimension — a line, or a single point. That's exactly the condition under which the matrix has no inverse, which is why "can't be undone" and "zero determinant" mean the same thing.
Transpose is the simplest operation on offer: swap rows and columns, row 1 becomes column 1, and so on — no arithmetic involved, just repositioning.