Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 1E from Chapter 1.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 1E
Chapter:
Problem:
Compute the products in Exercises 1-4 using (a) the definition, as in Example 1, and (b) the row-vector rule for computing Ax. If a product is undefined, explain why...
Step-by-Step Solution
Step 1
Given matrix and vector:
\[\begin{array}{*{20}{l}}{M = \left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]}\\{}\\{V = \left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]}\end{array}\]We have to determine the product of the matrix, M and the vector V by using the definition given in Example-1 and using the row vector Rule
Step 2: The definition for product in example-1
\[\begin{array}{l}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{{\bf{a}}_{\bf{1}}}}&{{{\bf{a}}_{\bf{2}}}}& \cdots &{{{\bf{a}}_{\bf{n}}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\\\A{\bf{x}} = {x_1}{{\bf{a}}_{\bf{1}}} + {x_2}{{\bf{a}}_{\bf{2}}} + \cdots + {x_n}{{\bf{a}}_{\bf{n}}}\end{array}\]The obvious condition for the product is that number of columns of A should be equal to number of entries in $\bf{x}$. For the given data number of columns of M are 2 which is not equal to number of elements in vector V (3). So the product is impossible.
ANSWER
The product is Impossible"
Given matrix and vector:
\[\begin{array}{*{20}{l}}{M = \left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]}\\{}\\{V = \left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]}\end{array}\]We have to determine the product of the matrix, M and the vector V by using the definition given in Example-1 and using the row vector Rule
Step 2: The definition for product in example-1
\[\begin{array}{l}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{{\bf{a}}_{\bf{1}}}}&{{{\bf{a}}_{\bf{2}}}}& \cdots &{{{\bf{a}}_{\bf{n}}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\\\A{\bf{x}} = {x_1}{{\bf{a}}_{\bf{1}}} + {x_2}{{\bf{a}}_{\bf{2}}} + \cdots + {x_n}{{\bf{a}}_{\bf{n}}}\end{array}\]The obvious condition for the product is that number of columns of A should be equal to number of entries in $\bf{x}$. For the given data number of columns of M are 2 which is not equal to number of elements in vector V (3). So the product is impossible.
ANSWER
The product is Impossible"