An identity matrix is a special type of square matrix where all the elements on the main diagonal (from the top-left to the bottom-right) are equal to 1, and all other elements are 0. The identity matrix is denoted as Iₙ, where n represents the number of rows (and columns) in the matrix. For any matrix A, multiplying it by the identity matrix (of appropriate size) leaves the matrix unchanged:

$I_n \times A = A \times I_n = A$

For example:

1. The 2x2 identity matrix: $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

2. The 3x3 identity matrix: $I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

In general, an identity matrix serves as the matrix equivalent of the number 1 in scalar multiplication.

Would you like further details or have any questions?

Related Questions:

1. How do you prove that multiplying a matrix by the identity matrix results in the same matrix?
2. What is the role of the identity matrix in matrix inversion?
3. How does the identity matrix relate to matrix transformations in geometry?
4. Can an identity matrix have a size other than square? Why or why not?
5. How do identity matrices play a role in linear systems of equations?

Tip:

When solving matrix equations, think of the identity matrix as the "do-nothing" operator, much like multiplying by 1 in algebra.