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Compute the following matrix vector products. show your work. if a product is undefined,explain why.

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Final Answer:

The matrix-vector products are as follows:

1.
\(AB\mathbf{v}\) is undefined.

2.
\(BA\mathbf{v}\) is defined.

Step-by-step explanation:

The product
\(AB\mathbf{v}\) is undefined because the number of columns in matrix (A) (denoted as
\(n_A\)) must be equal to the number of rows in matrix (B) (denoted as
\(m_B\))) for the product (AB) to be defined.

In this case, if (A) is an
\(m * n_A\) matrix and (B) is a
\(n_B * p\) matrix, the product (AB) is only defined if
\(n_A = n_B\). In the given question, the vector
\(\mathbf{v}\) is of size
\(n_B * 1\). However, since (A) and (B) have different numbers of columns
(\(n_A\) and
\(n_B\)), the product
\(AB\mathbf{v}\) cannot be computed.

On the other hand, the product
\(BA\mathbf{v}\) is defined. In this case, matrix (B) has
\(n_B\) columns, and matrix (A) has
\(n_A\) columns. Therefore, the product (BA) is defined, resulting in a matrix of size
\(m_B * n_A\). Multiplying this matrix by the vector
\(\mathbf{v}\) (of size \(n_A * 1\)) yields a vector of size
\(m_B * 1\), and the product is well-defined.

In summary, the product
\(AB\mathbf{v}\) is undefined due to a mismatch in the number of columns between matrices (A) and (B), while the product
\(BA\mathbf{v}\) is defined since the matrices (B) and (A) satisfy the necessary conditions for multiplication.

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