State the transpose of \(I_{3}\).

An identity matrix is a square matrix that is a key concept in linear algebra. It plays the role of the number 1 in the matrix world; just like multiplying any number by 1 gives the original number, multiplying any matrix by the identity matrix gives the original matrix back. An identity matrix is denoted as 'I' and has dimensions of n x n. It is filled with 0s except for the main diagonal, which consists of 1s. This can be visually represented for a 3x3 matrix as:
\[ I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]
The number subscript refers to the size of the matrix, so for instance, for a 2x2 matrix, the identity matrix would look like \( I_2 \) and have fewer elements but still follow the same pattern. The identity matrix is essential when discussing matrix inverse, solving systems of linear equations, and understanding linear transformations.

The transpose of a matrix is a fundamental operation in linear algebra where the rows of a matrix are turned into columns and vice versa. Mathematically, this is denoted as \( A^T \) for any given matrix A, and it effectively flips the matrix over its diagonal. For example, if we have a 2x3 matrix:
\[ A = \begin{bmatrix} a & b & c \ d & e & f \end{bmatrix} \],
its transpose would be a 3x2 matrix:
\[ A^T = \begin{bmatrix} a & d \ b & e \ c & f \end{bmatrix} \].

Properties of the Transpose:

• \( (A^T)^T = A \): Transposing a matrix twice will yield the original matrix.
• \( (AB)^T = B^T A^T \): The transpose of the product of two matrices is the product of their transposes in reverse order.
• For square matrices, the transpose of the matrix is itself if it is symmetric, meaning that \( A = A^T \).

As illustrated in the original exercise, transposing an identity matrix leaves it unchanged because it is a symmetric matrix.

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