Identity, Inverse and Transpose
Three matrices built from one. The identity does nothing, the inverse undoes, and the transpose — which people reach for as if it were an undo — does something else entirely.
The Matrix A
Do And Undo
the transpose is not the inverse — unless A is orthogonal
Press Apply A. The shape moves; press the second button to try to put it back.
There And Back
—The faint outline is where the shape started. Green is where it is now.
A⁻¹
Every entry is divided by the determinant, which is exactly why a determinant of zero leaves no inverse to write down.
A A⁻¹ And Aᵀ
Identity, Inverse and Transpose
These three matrix operations describe doing nothing, undoing a map, and exchanging rows with columns.
Inverse undoes a transformation
For square A, its inverse A⁻¹ satisfies A⁻¹A = AA⁻¹ = I. It exists only when A is nonsingular. If A sends two distinct inputs to the same output, no operation can recover which input was used. Numerically, solving Ax = b with a factorisation is usually preferable to explicitly computing A⁻¹.
Transpose swaps rows and columns
The transpose Aᵀ is formed by exchanging rows and columns. It reverses product order: (AB)ᵀ = BᵀAᵀ. For an orthogonal matrix Q, QᵀQ = I, so Qᵀ = Q⁻¹. This is why rotations and orthonormal coordinate changes are easy to undo. See orthogonality, matrix multiplication, and projections.
A useful check: for A = [[1, 2], [3, 4]], Aᵀ = [[1, 3], [2, 4]], while det(A) = −2, so this particular matrix has an inverse. A transpose always exists; an inverse does not.
Undo a map by solving a system
For A=[[1,2],[3,4]], the determinant is −2, so the inverse exists and is A⁻¹=[[-2,1],[1.5,−0.5]]. If Ax=b and b=(5,11), solving gives x=(1,2); applying A⁻¹ to b recovers the input. If A is singular, two different inputs may have the same output, so there is no single inverse that can undo the transformation for every vector. Aᵀ always exists, but A⁻¹ may not; transpose and inverse are not interchangeable.
For orthogonal Q, columns are unit length and mutually perpendicular, so QᵀQ=I and Qᵀ=Q⁻¹. A rotation is therefore undone by reversing its direction. The identity is the multiplicative neutral element, analogous to 1 for numbers. In numerical computing, solve Ax=b using a factorisation rather than explicitly forming A⁻¹: it is usually faster and more stable. Check the equation and matrix dimensions; right and left multiplication can encode different operations.
Try this above
Compare a matrix with its transpose, then use the inverse control to map a transformed grid back. Try a singular matrix to see why it cannot be undone.
A quick inverse check is to multiply both ways and see whether the result is the identity. For the example, A⁻¹A=AA⁻¹=I, while AᵀA is not I, so the transpose is not the inverse. The transpose does preserve the dimensions of a square matrix but swaps row and column roles; it is especially useful in dot products and quadratic forms. For rectangular A, the transpose is still well-defined but has no ordinary two-sided inverse. Least-squares methods address rectangular systems without pretending a standard inverse exists.
When A is symmetric positive-definite, it has an inverse and a Cholesky factor, but computing an inverse is still usually unnecessary for solving Ax=b. If A is orthogonal, the transpose is its inverse; this special case follows from preserving lengths and angles. For a general matrix, use the transpose to move between row and column representations, not as an undo operation. This distinction appears throughout least squares, covariance calculations, and neural-network gradients.
Sources and next steps
- MIT OpenCourseWare: Linear Algebra covers inverses, transposes, and orthogonal matrices.
- Review determinants to connect invertibility with volume collapse.
Overview
Identity leaves a vector unchanged
The identity matrix has ones on its diagonal and zeros elsewhere. Multiplying by it changes nothing: I x = x and AI = IA = A whenever the products are defined. It plays the role of 1 in matrix multiplication.