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 9E from Chapter 2.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 9E
Chapter:
Problem:
In Exercises 9 and 10, mark each statement True or False. Justify each answer. a. In order for a matrix B to be the inverse of A, the equations AB = I and BA =
Step-by-Step Solution
Given Information
we are given with some statements, that we have to find whether they are True or False
Step 1: Part (a)
Statement: In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.
The Statement is true. By the definition of invertible matrix, if a matrix B is inverse of matrix A, then \[AB = BA = I\]Therefore
TRUE
Step 2: Part (b)
Statement: If A and B are $n \times n$ and invertible, then $A{^-1}B^{-1}$ is theinverse of AB.
The Statement is False. if A and B are two invertible matrices, then inverse of their product is given by\[{\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^ - 1 }\]Therefore
FALSE
Step 3: Part (c)
Statement: If $A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$ and $ab - cd \ne 0$; then A is invertible.
The Statement is False. The necessary condition for an invertible matrix is that $cb - ad \ne 0$
Therefore
FALSE
Step 4: Part (d)
Statement: If A is an invertible $n \times n$ matrix, then the equation$Ax= b$ is consistent for each b in $R^n$
The Statement is True. By the theorem, we know that if A is an invertible matrix, than the system of equation $Ax=b $must have a unique solution so $Ax=b $ is consistent for each $b$ in $R^n$
Therefore
TRUE
Step 5: Part (e)
Statement: Each elementary matrix is invertible.
The Statement is True. An elementary matrix obtained by applying a single row operation on the identify matrix and the inverse of E is an elementary matrix of the transformation of matrix into identity matrix.
Therefore
TRUE
we are given with some statements, that we have to find whether they are True or False
Step 1: Part (a)
Statement: In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.
The Statement is true. By the definition of invertible matrix, if a matrix B is inverse of matrix A, then \[AB = BA = I\]Therefore
TRUE
Step 2: Part (b)
Statement: If A and B are $n \times n$ and invertible, then $A{^-1}B^{-1}$ is theinverse of AB.
The Statement is False. if A and B are two invertible matrices, then inverse of their product is given by\[{\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^ - 1 }\]Therefore
FALSE
Step 3: Part (c)
Statement: If $A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$ and $ab - cd \ne 0$; then A is invertible.
The Statement is False. The necessary condition for an invertible matrix is that $cb - ad \ne 0$
Therefore
FALSE
Step 4: Part (d)
Statement: If A is an invertible $n \times n$ matrix, then the equation$Ax= b$ is consistent for each b in $R^n$
The Statement is True. By the theorem, we know that if A is an invertible matrix, than the system of equation $Ax=b $must have a unique solution so $Ax=b $ is consistent for each $b$ in $R^n$
Therefore
TRUE
Step 5: Part (e)
Statement: Each elementary matrix is invertible.
The Statement is True. An elementary matrix obtained by applying a single row operation on the identify matrix and the inverse of E is an elementary matrix of the transformation of matrix into identity matrix.
Therefore
TRUE