The inverse of a matrix undoes that matrix, and their product is the identity. If the determinant is 0, there is no inverse.
M is a shear. Slide to 1 to apply M. Slide on to 2 to apply the inverse. The F returns.
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Read M, then M inverse, then the product. The product is the identity when an inverse exists. If the determinant is 0, there is no inverse. Lesson 06 squashes a shape onto a line. Lesson 09 flattens a house onto a plane. The lost direction does not come back.
M. Read each row from left to right.
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M⁻¹. This is the inverse of M.
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M times M⁻¹. This product is the identity when the inverse exists.
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Where this is used
A CAD program stores the placement of a part as a matrix. The inverse puts points back into the part's own axes. A mechanical designer uses that after a part has been moved.
A game camera uses a matrix to see the world from the camera. The inverse carries a point from the camera back into the world. A graphics programmer sets that up.
In a mathematics class, the Hill cipher is a historical example. Letters are encrypted by a matrix and decrypted by its inverse. It is a teaching cipher from 1929, not a modern one.
Common mistakes
Not every matrix has an inverse. A flatten has none, because the determinant is 0 and the lost direction does not come back.
The inverse is not the transpose, and it is not the matrix with the sign of every entry flipped.
When an inverse exists, the product either way is the identity. The slider on this page applies the matrix first, then the inverse.
The Hill cipher decrypts with an inverse. It is a teaching cipher from 1929, not a modern one.