Identity Matrix

The identity matrix is a square matrix such that all the entries in the main diagonal are 1, and the rest of the entries are all 0. The identity matrix is denoted by \( I \) in general. It is also denoted by \( I_{n} \), where \( n \) is the number of rows in the matrix. The \( 4 \times 4 \) identity matrix \( I_{4} \) is shown below:

\[ I_{4} = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]. \]

The unique property of the identity matrix is that multiplying it with another matrix (of a suitable order) does not change the matrix:

\[ IA = A = AI. \]

Multiplying any matrix by its inverse yields the identity matrix:

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

The identity matrix is a/an

• symmetric matrix
• diagonal matrix
• scalar matrix
• upper triangular matrix
• lower triangular matrix
• idempotent matrix
• involutory matrix
• orthogonal matrix .