Identity matrix

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An identity matrix or unit of order n is a square matrix where all its elements are zeros minus the elements on the main diagonal that are ones.

In other words, an identity matrix has only ones on the main diagonal and all other elements of the matrix with zeros. Furthermore, the identity matrix is ​​recognized as having a square shape since it is a square matrix.

Matrix operations

Representation of an identity matrix

Examples of identity matrices

We can create infinite combinations of unit matrices as long as we respect the condition of being a square matrix: having the same number of rows (n) and columns (m).

Properties

When we carry out operations with the unit matrix we should not get nervous. We must think of the identity matrix as the number one.

Number 1

  • When we multiply any other number by one we are left with the same number (neutrality). Given a constant z or any scalar:

Neutral effect of multiplying a constant with the number one.
  • If we do the inverse of the number one, we will get the same number one (invertible).

Neutral effect of doing the inverse of the number one.
  • When we raise the number one h units, we will always have the number one (idempotency).

Neutral effect of raising the number one to a constant h.

Identity matrix

  • Neutrality. When the unit matrix participates in a multiplication of matrices, it is called a neutral product. Given any matrix Z:

Neutral effect of multiplying the identity matrix by any matrix.
  • Reversible. The inverse matrix of the unit matrix is ​​the identity matrix:

Neutral effect of inverting the identity matrix.
  • Idempotency. The raised inverse matrix h units (natural number) is still the unit matrix:

Neutral effect of raising the identity matrix by h units.

Procedure to identify an identity matrix

  1. The matrix has to be a square matrix.
  2. The matrix must have ones on the main diagonal and zeros in the other positions.

Applications

The identity matrix participates as many times as the number one participates in algebra. For example, when we multiply any matrix with its inverse matrix, we will obtain the unit matrix.

Theoretical example

Are the following matrices identity matrices?

Examples of identity matrices and non-identity matrices.

Matrix IA:

  • Square matrix.
  • Non-identity matrix: on the main diagonal there is a number other than one and in the other positions there is a number other than zero.

Matrix IB:

  • Not square matrix.
  • No identity matrix.

IC matrix:

  • Not square matrix.
  • No identity matrix.

Matrix ID:

  • Square matrix.
  • Identity matrix: on the main diagonal there are ones and in the other positions there are zeros.

IE matrix:

  • Square matrix.
  • No identity matrix: although in the other positions there are zeros, in the main diagonal there is a number other than one.
Determinant of a matrix

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