Add, subtract, multiply or invert. Determinants for both matrices are shown throughout.
Matrix A starts as the identity matrix — 1s on the diagonal, 0s elsewhere — and matrix B starts at [[2,1],[1,2]]. Multiplying them, A × B, returns exactly B unchanged: [[2,1],[1,2]]. That is the defining property of an identity matrix made visible rather than just stated — multiplying by it is the matrix equivalent of multiplying an ordinary number by 1. det(A) = 1 confirms it: the identity scales area by a factor of exactly one, meaning it doesn't transform anything at all.
det(B) = 2×2 − 1×1 = 3, so B is invertible: swap the diagonal, negate the off-diagonal, divide by 3, giving B⁻¹ = [[2/3, −1/3], [−1/3, 2/3]]. Multiplying B by that inverse — switch the operation to A × B with A's identity replaced by this result — returns the identity matrix again, confirming the inverse actually undoes B rather than just looking like a plausible answer.
The 2×2 shortcut — swap, negate, divide — has no 3×3 equivalent; there's no analogous "swap the corners" trick for a bigger grid. Switching to a 3×3 here instead uses cofactor expansion: build a grid of signed 2×2 sub-determinants from whatever remains after deleting each row and column in turn, then divide the whole result by the full matrix's determinant. It's a completely different algorithm, not a bigger version of the same one, which is why matrix inversion gets noticeably harder to do by hand past 2×2.
A transformation matrix is exactly how graphics software rotates, scales, and repositions shapes on a 2D or 3D coordinate grid — the same (x, y) points the Coordinate Distance Calculator works with directly, just moved by a matrix multiplication instead of typed in by hand. And matrix entries are ordinary numbers with no special size limit of their own — for a determinant or product that overflows what floating point can hold exactly, the Big Number Calculator covers the arithmetic underneath at any scale, and for arithmetic that never involves a grid of numbers in the first place, the Scientific Calculator handles everything else.
For ordinary numbers 3 × 5 and 5 × 3 are the same. For matrices, A × B is generally not B × A. Swapping the order usually gives a different matrix, and sometimes an invalid operation.
This is not a quirk — it reflects what matrices represent. A matrix is a transformation, and rotating then stretching is genuinely not the same as stretching then rotating. The order of operations is part of the meaning.
Multiplication also works by rows into columns: entry (i, j) of the product is the dot product of row i of A with column j of B. Addition, by contrast, is simply element by element.
The determinant is a single number summarising the whole matrix, and it answers one question immediately: is this matrix invertible? If the determinant is zero, it is not.
Geometrically it is a scale factor. A 2×2 matrix with determinant 10 multiplies areas by 10; a 3×3 with determinant 10 multiplies volumes by 10. A determinant of zero means the transformation collapses space into a lower dimension — squashing a plane onto a line — and you cannot undo that, which is exactly why the inverse fails to exist.
A negative determinant means the transformation flips orientation, turning a shape into its mirror image.
A × A⁻¹ gives the identity matrix — 1s down the diagonal, 0s elsewhere — which is the matrix equivalent of multiplying by one. For a 2×2 the inverse is straightforward: swap the diagonal entries, negate the off-diagonal ones, and divide everything by the determinant.
Inverses are how simultaneous equations get solved in one step. Writing a system as Ax = b, the solution is x = A⁻¹b. A determinant of zero here means the system has either no solution or infinitely many — never exactly one.
Because matrices represent transformations, and the order you apply transformations changes the outcome. Rotating then stretching is not the same as stretching then rotating. Matrix multiplication is associative but not commutative.
The matrix is singular and has no inverse. Geometrically the transformation collapses space into a lower dimension — a plane onto a line, say — and that cannot be reversed. For simultaneous equations it means there is no unique solution.
The factor by which the transformation scales area in 2D or volume in 3D. A determinant of 10 multiplies areas tenfold; a negative determinant means the orientation is flipped, producing a mirror image.
A system written as Ax = b has the solution x = A⁻¹b. Instead of eliminating variables one at a time, you invert the coefficient matrix once and multiply. It only works when the determinant is non-zero.
For determinants of larger systems and general computation, the Scientific Calculator handles the underlying arithmetic. Coordinate transformations connect to the Coordinate Distance Calculator and Slope Calculator.