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 11E from Chapter 2.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 11E
Chapter:
Problem:
In Exercises 11 and 12, mark each statement True or False. Justify each answer. a. If with A1 and A2 the same sizes as B1 and B2, respectively, then b. If then the partitions of A and B are conformable for block multiplication.
Step-by-Step Solution
Given Information
We are given with some statements that we have to check if thy are True or False.
Step-1: (a)
Given following matrices; $$A = \left[ A _ { 1 } , A _ { 2 } \right]$ and $B = \left[ B _ { 1 } , B _ { 2 } \right]$$ The matrices $A _ { 1 }$ and $A _ { 2 }$ have the same sizes as that of matrices $B _ { 1 }$ and $B _ { 2 }$
The matrix addition is possible only if both the matrices are partitioned in exactly the same way. Both the matrices A and B have two column partitions and one row partition.
Therefore, the matrices A and B can be added and their sum is the partitioned matrix obtained by adding the respective block matrices. Therefore,
The statement is True
Step-2: (b)
We are given with following matrices: \[ A = \left[ \begin{array} { l l } { A _ { 11 } } & { A _ { 12 } } \\ { A _ { 21 } } & { A _ { 22 } } \end{array} \right] \text { and } B = \left[ \begin{array} { l } { B _ { 1 } } \\ { B _ { 2 } } \end{array} \right] \] If the column partitions of the first matrix are same as the row partitions of the second matrix., then block multiplication is possible.
The matrix A has two column and two row partitions.
Matrix B has two row and one column partition. Therefore, the block multiplication is not possible.
Therefore,
The statement is False
We are given with some statements that we have to check if thy are True or False.
Step-1: (a)
Given following matrices; $$A = \left[ A _ { 1 } , A _ { 2 } \right]$ and $B = \left[ B _ { 1 } , B _ { 2 } \right]$$ The matrices $A _ { 1 }$ and $A _ { 2 }$ have the same sizes as that of matrices $B _ { 1 }$ and $B _ { 2 }$
The matrix addition is possible only if both the matrices are partitioned in exactly the same way. Both the matrices A and B have two column partitions and one row partition.
Therefore, the matrices A and B can be added and their sum is the partitioned matrix obtained by adding the respective block matrices. Therefore,
The statement is True
Step-2: (b)
We are given with following matrices: \[ A = \left[ \begin{array} { l l } { A _ { 11 } } & { A _ { 12 } } \\ { A _ { 21 } } & { A _ { 22 } } \end{array} \right] \text { and } B = \left[ \begin{array} { l } { B _ { 1 } } \\ { B _ { 2 } } \end{array} \right] \] If the column partitions of the first matrix are same as the row partitions of the second matrix., then block multiplication is possible.
The matrix A has two column and two row partitions.
Matrix B has two row and one column partition. Therefore, the block multiplication is not possible.
Therefore,
The statement is False