Editorial StaffAn identity matrix is a special type of square matrix that has ones on its main diagonal (from the top left to the bottom right) and zeros everywhere else. It is denoted by the symbol $I$ or $I_n$, where $n$ represents the size of the matrix.
For example, the $2\times 2$ identity matrix, denoted as $I_2$, is defined as:
\[
I_2 =
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
\]
Similarly, the $3\times 3$ identity matrix, denoted as $I_3$, is defined as:
\[
I_3 =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\]
Properties
1. The identity matrix has some interesting properties. When multiplied with any matrix of compatible dimensions, it leaves the matrix unchanged. That is, for any $m\times n$ matrix $A$, we have:
\[
A \cdot I = I \cdot A = A
\]
where $A \cdot I$ represents the matrix multiplication of $A$ and $I$, and $I \cdot A$ represents the matrix multiplication of $I$ and $A$. This property makes the identity matrix act like the number 1 in scalar multiplication.
2. The identity matrix is also invertible. The inverse of an $n\times n$ identity matrix $I_n$ is itself:
\[
I_n^{-1} = I_n
\]
3. This means that multiplying the identity matrix by its inverse gives the original matrix:
\[
I_n \cdot I_n^{-1} = I_n^{-1} \cdot I_n = I_n
\]
The identity matrix plays a crucial role in linear algebra, matrix operations, and solving systems of linear equations. It serves as a fundamental building block for many matrix-related calculations.
