What is a unit matrix?

A unit matrix, also known as an identity matrix, is a square matrix with ones on the main diagonal and zeros elsewhere. The main diagonal of a matrix refers to the elements that run from the top left to the bottom right. In mathematical notation, an n x n unit matrix is represented as Iₙ or simply I, where n represents the size or order of the matrix.

The elements of an n x n unit matrix can be denoted as follows:

Iₙ = | 1 0 0 ... 0 |
| 0 1 0 ... 0 |
| 0 0 1 ... 0 |
| ... ... ... ... ... |
| 0 0 0 ... 1 |

For example, the 3 x 3 unit matrix would be represented as:

I₃ = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |

The unit matrix is called such because it behaves like the number 1 in matrix multiplication. When a matrix A is multiplied by the unit matrix I, the result is the matrix A itself. In other words, for any matrix A of appropriate size:

A x I = I x A = A

The unit matrix plays a fundamental role in linear algebra and matrix operations, serving as the multiplicative identity element for matrices. It is often used in various mathematical operations, including matrix inverses, solving systems of linear equations, and defining transformations in linear algebra.