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 15E from Chapter 2.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 15E
Chapter:
Problem:
Exercises 15 and 16 concern arbitrary matrices A, B, and C for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. a. If A and B are
Step-by-Step Solution
Given Information
We have to marks the given statements as True or False and justify for the same.
Step 1: (a)
Statement: If A and B are $2 \times 2$ with columns $a_1; a_2$ and $b_1, b2$, respectively, then $AB = \left[ {{{\bf{a}}_{\bf{1}}}{{\bf{b}}_{\bf{1}}}\,\,\,\,{{\bf{a}}_{\bf{2}}}{{\bf{b}}_{\bf{2}}}} \right]$.
The Statement is False! For the product of two matrices, we need to have rows of A ($a_1; a_2$) and columns B ($b_1, b2$), then we can find the product as: \[AB = \left[ {{{\bf{a}}_{\bf{1}}}{{\bf{b}}_{\bf{1}}}\,\,\,\,{{\bf{a}}_{\bf{2}}}{{\bf{b}}_{\bf{2}}}} \right]\] However, we are only provided with columns of the matrices.
Therefore FALSE
Step 2: (b)
Statement: Each column of AB is a linear combination of the columnsof B using weights from the corresponding column of A..
The Statement is False! Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.
Therefore FALSE
Step 3: (c)
Statement: $AB + AC = A\left( {B + C} \right)$.
The Statement is TRUE! By the left distributive property, the statement is True
Therefore TRUE
Step 4: (d)
Statement: ${A^T} + {B^T} = {(A + B)^T}$.
The Statement is TRUE! By the properties of the transposition, the transpose of sum of matrices is sum of transposes of those matrices.
Therefore TRUE
Step 5: (e)
Statement: The transpose of a product of matrices equals the productof their transposes in the same order..
The Statement is TRUE! By the properties of the transposition, the transpose of the product matrices equals the product of their transposes in reverse order.
Therefore TRUE
We have to marks the given statements as True or False and justify for the same.
Step 1: (a)
Statement: If A and B are $2 \times 2$ with columns $a_1; a_2$ and $b_1, b2$, respectively, then $AB = \left[ {{{\bf{a}}_{\bf{1}}}{{\bf{b}}_{\bf{1}}}\,\,\,\,{{\bf{a}}_{\bf{2}}}{{\bf{b}}_{\bf{2}}}} \right]$.
The Statement is False! For the product of two matrices, we need to have rows of A ($a_1; a_2$) and columns B ($b_1, b2$), then we can find the product as: \[AB = \left[ {{{\bf{a}}_{\bf{1}}}{{\bf{b}}_{\bf{1}}}\,\,\,\,{{\bf{a}}_{\bf{2}}}{{\bf{b}}_{\bf{2}}}} \right]\] However, we are only provided with columns of the matrices.
Therefore FALSE
Step 2: (b)
Statement: Each column of AB is a linear combination of the columnsof B using weights from the corresponding column of A..
The Statement is False! Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.
Therefore FALSE
Step 3: (c)
Statement: $AB + AC = A\left( {B + C} \right)$.
The Statement is TRUE! By the left distributive property, the statement is True
Therefore TRUE
Step 4: (d)
Statement: ${A^T} + {B^T} = {(A + B)^T}$.
The Statement is TRUE! By the properties of the transposition, the transpose of sum of matrices is sum of transposes of those matrices.
Therefore TRUE
Step 5: (e)
Statement: The transpose of a product of matrices equals the productof their transposes in the same order..
The Statement is TRUE! By the properties of the transposition, the transpose of the product matrices equals the product of their transposes in reverse order.
Therefore TRUE