Identity Matrix

An identity matrix, denoted as \( I_n \), is a square matrix in which all the diagonal elements are 1s, and all the other elements are 0s. It serves as the multiplicative identity in matrix algebra, meaning that multiplying any matrix by the identity matrix results in the original matrix.

Mathematically, an identity matrix of order \( n \times n \) is represented as:

\[ I_n = \begin{bmatrix} 1 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{bmatrix} \]

Examples of Identity Matrices

  • The 2×2 identity matrix is: \[ I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]
  • The 3×3 identity matrix is: \[ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
  • The 4×4 identity matrix is: \[ I_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

Properties of Identity Matrix

1 Multiplicative Identity Property

Multiplying any matrix \( A \) by the identity matrix \( I \) results in the original matrix:

\[ A \times I = I \times A = A \]

Example: Consider the matrix:

\[ A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \]

Multiplying \( A \) by the identity matrix \( I_2 \):

\[ I_2 \times A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \]

The result is the original matrix \( A \).

2 Identity Matrix is Its Own Inverse

The inverse of the identity matrix is the identity matrix itself:

\[ I^{-1} = I \]

Example: The inverse of \( I_3 \) is:

\[ I_3^{-1} = I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

3 Identity Matrix in Matrix Multiplication

The identity matrix behaves like 1 in scalar multiplication:

\[ I_n \times A = A \times I_n = A \]

Example: If:

\[ B = \begin{bmatrix} 7 & -2 \\ 5 & 6 \end{bmatrix} \]

Then:

\[ I_2 \times B = B \times I_2 = B \]

4 Identity Matrix and Determinant

The determinant of an identity matrix is always 1:

\[ \det(I_n) = 1 \]

Example: For \( I_3 \):

\[ \det \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = 1 \]

5 Identity Matrix in Linear Transformations

The identity matrix represents the transformation that leaves vectors unchanged:

\[ I_n \times v = v \]

Example: If:

\[ v = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \]

Then:

\[ I_2 \times v = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \]