Question

Compute the products in Exercises 1–4 using (a) the definition, as

in Example 1, and (b) the row–vector rule for computing \(A{\bf{x}}\). If a product is undefined, explain why.

1. \(\left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]\)

Short answer

The product is not defined because the number of columns in the matrix does not match the number of entries in the vector.

Step-by-step solution

Write the condition for the product of a vector and a matrix

According to the definition, the weights in a linear combination of matrix A columns are represented by the entries in vector x.

Also, the product by using the row-vector rule is defined as shown below:

\(\begin{array}{c}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\ = {x_1}{a_1} + {x_2}{a_2} + \cdots + {x_n}{a_n}\end{array}\)

The number of columns in matrix \(A\) should be equal to the number of entries in vector x so that \(A{\bf{x}}\) can be defined.

Obtain the number of columns in matrix A

Consider matrix \(A = \left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]\).

It is observed that the number of columns in matrix \(A\) is 2.

Obtain the number of entries in vector x

Consider matrix \(x = \left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]\).

It is observed that the number of entries in vector x is 3.

Check if \(Ax\) is defined or not

Since the number of columns in matrix \(A\) is not equal to the number of entries in vector x; so \(A{\bf{x}}\) cannot be defined.

Therefore, the product of \(\left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]\) is undefined.

Hence, the product is not defined.