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Matrix multiplication is a fundamental operation in linear algebra. It involves finding the product of two matrices, often denoted as \( C = AB \). To multiply two matrices, it is important to follow specific rules:

• Dimensions must align: The number of columns in the first matrix \( A \) must equal the number of rows in the second matrix \( B \). If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times p \) matrix, their product \( AB \) will be an \( m \times p \) matrix.
• Entry calculation: Each entry \( c_{ij} \) of the resulting matrix \( C \) is computed by taking the dot product of the \( i \text{th} \) row of \( A \) and the \( j \text{th} \) column of \( B \).

In the example given, we multiplied matrices \( A \) and \( B \), where \( A = \begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix} \) and \( B = \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix} \). The resulting product \( AB \) was the zero matrix \( \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \), as each entry was the result of multiplying a zero with corresponding entries of \( B \) or \( A \).
Matrix multiplication is associative and distributive but not commutative, meaning \( AB eq BA \) in general.

In linear algebra, a matrix is termed as singular if it does not have an inverse. For a square matrix \( A \), this means that there is no matrix \( B \) such that \( AB = BA = I \), where \( I \) is the identity matrix.

• Zero determinant: A matrix is singular when its determinant is zero, indicating that it cannot be inverted.
• Non-full rank: Singular matrices do not have full rank, meaning not all of their rows or columns are linearly independent.

In the exercise, both matrix \( A \) and matrix \( B \) are singular, as seen with their diagonal structures. They do not possess full rank due to zero rows or columns. Their singularity plays a key role in the exercise, as it highlights why the pseudoinverse was calculated as opposed to a regular inverse. This concept is crucial when working with non-invertible matrices in various applications such as solving system equations and optimization.

Diagonal matrices are a simple yet powerful type of matrix characterized by having non-zero entries only on their main diagonal. All other positions outside this diagonal are zeros. This unique structure gives diagonal matrices several useful properties:

• Easier calculations: Operations like addition, subtraction, and finding powers of the matrix become simpler due to their zero off-diagonal entries.
• Diagonal pseudoinverses: The pseudoinverse of a diagonal matrix is particularly straightforward. Each non-zero diagonal element is inverted while zero elements remain unchanged, as seen with matrices \( A \) and \( B \) in the exercise.
• Eigensystem simplification: The eigenvalues of a diagonal matrix are simply the entries on the main diagonal, and their eigenvectors correspond to the standard basis vectors.

In this exercise, both matrices \( A \) and \( B \) are diagonal, which simplifies the computation of their pseudoinverses. This makes the calculations efficient and demonstrates why diagonal matrices are frequently utilized in theoretical and applied mathematics problems.