Question

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

Answer 1

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.