Special Matrices

3. An identity matrix is a square matrix that has $1$s on its main diagonal, and $0$s elsewhere.
   • We use the letter $I$ to denote an identity matrix. For example, the identity matrix of size $3 \times 3$ is $I = I_{3 \times 3} = I_3$.
     • For the identity $I_n = [i_{j,k}]$, we have $i_{j,j} = 1$, where $1 \le j \le n$, and $i_{j,k} = 0$, where $1 \le j, k \le n$ but $j \neq k$.
   
   Examples:

   19. $I_1 = \begin{bmatrix} 1 \end{bmatrix}$

   23. $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

   24. $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

   6. $I_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

   25. $I_5 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{bmatrix}$