Matrix arithmetic, 2×2 and 3×3.

Multiplication is not commutative

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.

What the determinant tells you

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.

The inverse undoes the transformation

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.

Why is A × B different from B × A?

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.

What does a determinant of zero mean?

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.

What does the determinant actually measure?

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.

How is the inverse used to solve equations?

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.