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 2E from Chapter 2.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 2E
Chapter:
Problem:
In Exercises, compute each matrix sum or product if it is defined. If an expression is undefined, explain why...
Step-by-Step Solution
Step 1
Given Matrices\[\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{array}} \right]\\\\B = \left[ {\begin{array}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{array}} \right]\\\\C = \left[ {\begin{array}{*{20}{c}}3&5\\{ - 2}&1\end{array}} \right]\\\\D = \left[ {\begin{array}{*{20}{c}}3&5\\{ - 1}&4\end{array}} \right]\\\\E = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\end{array}} \right]\end{array}\]We have to find the following entities :\[\begin{array}{l}\left( a \right)\,\,A + 2B\\\\\left( b \right)\,\,3C - E\\\\\left( c \right)\,\,CB\\\\\left( d \right)\,EB\end{array}\]
Step 2: (a) A + 2B
This is just an multiplication of scalar with a matrix and addition with another matrix. Each element of the matrix B will be multiplied with the scalar, 2 and added to corresponding element of A (since both matrices have same dimension).\[\begin{array}{l}A + 2B = \left[ {\begin{array}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{array}} \right] + 2\left[ {\begin{array}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{14}&{ - 10}&2\\2&{ - 8}&{ - 6}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{2 + 14}&{0 - 10}&{ - 1 + 2}\\{4 + 2}&{ - 5 - 8}&{2 - 6}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{array}} \right]\end{array}\]
Step 3: (b) 3C - E
The operations involved are multiplication of scalar with a matrix and then matrix addition. However, since dimensions of E and C are not same, the matrix additon is not possible.
Step 4: (c) CB
The operations involve multiplication of matrix C with B. As number of columns of C are 2 which is equal to number of rows of B, the operation is possible. \[\begin{array}{l}CB = \left[ {\begin{array}{*{20}{c}}1&2\\{ - 2}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{1 \times 7 + 2 \times 1}&{1 \times ( - 5) + 2 \times ( - 4)}&{1 \times 1 + 2 \times ( - 3)}\\{ - 2 \times 7 + 1 \times 1}&{ - 2( - 5) + 1 \times ( - 4)}&{ - 2 \times 1 + 1 \times ( - 3)}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{7 + 2}&{ - 5 - 8}&{1 + ( - 6)}\\{ - 14 + 1}&{ + 10 - 4}&{ - 2 - 3}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{array}} \right]\end{array}\]
Step 5: EB
The operations involves multiplication of matrix E with B. As number of columns of E is 1 which is not equal to number of rows of B (2), hence the operation is not possible.
ANSWERS
\[\begin{array}{l}A + 2B = \left[ {\begin{array}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{array}} \right]\\\\3C - E:\,\,{\rm{NOT}}\,{\rm{POSSIBLE}}\\\\CB = \left[ {\begin{array}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{array}} \right]\\\\EB:\,\,{\rm{NOT}}\,{\rm{POSSIBLE}}\end{array}\]
Given Matrices\[\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{array}} \right]\\\\B = \left[ {\begin{array}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{array}} \right]\\\\C = \left[ {\begin{array}{*{20}{c}}3&5\\{ - 2}&1\end{array}} \right]\\\\D = \left[ {\begin{array}{*{20}{c}}3&5\\{ - 1}&4\end{array}} \right]\\\\E = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\end{array}} \right]\end{array}\]We have to find the following entities :\[\begin{array}{l}\left( a \right)\,\,A + 2B\\\\\left( b \right)\,\,3C - E\\\\\left( c \right)\,\,CB\\\\\left( d \right)\,EB\end{array}\]
Step 2: (a) A + 2B
This is just an multiplication of scalar with a matrix and addition with another matrix. Each element of the matrix B will be multiplied with the scalar, 2 and added to corresponding element of A (since both matrices have same dimension).\[\begin{array}{l}A + 2B = \left[ {\begin{array}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{array}} \right] + 2\left[ {\begin{array}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{14}&{ - 10}&2\\2&{ - 8}&{ - 6}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{2 + 14}&{0 - 10}&{ - 1 + 2}\\{4 + 2}&{ - 5 - 8}&{2 - 6}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{array}} \right]\end{array}\]
Step 3: (b) 3C - E
The operations involved are multiplication of scalar with a matrix and then matrix addition. However, since dimensions of E and C are not same, the matrix additon is not possible.
Step 4: (c) CB
The operations involve multiplication of matrix C with B. As number of columns of C are 2 which is equal to number of rows of B, the operation is possible. \[\begin{array}{l}CB = \left[ {\begin{array}{*{20}{c}}1&2\\{ - 2}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{1 \times 7 + 2 \times 1}&{1 \times ( - 5) + 2 \times ( - 4)}&{1 \times 1 + 2 \times ( - 3)}\\{ - 2 \times 7 + 1 \times 1}&{ - 2( - 5) + 1 \times ( - 4)}&{ - 2 \times 1 + 1 \times ( - 3)}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{7 + 2}&{ - 5 - 8}&{1 + ( - 6)}\\{ - 14 + 1}&{ + 10 - 4}&{ - 2 - 3}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{array}} \right]\end{array}\]
Step 5: EB
The operations involves multiplication of matrix E with B. As number of columns of E is 1 which is not equal to number of rows of B (2), hence the operation is not possible.
ANSWERS
\[\begin{array}{l}A + 2B = \left[ {\begin{array}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{array}} \right]\\\\3C - E:\,\,{\rm{NOT}}\,{\rm{POSSIBLE}}\\\\CB = \left[ {\begin{array}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{array}} \right]\\\\EB:\,\,{\rm{NOT}}\,{\rm{POSSIBLE}}\end{array}\]