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 10E from Chapter 2.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 10E
Chapter:
Problem:
In Exercises, mark each statement True or False. Justify each answer.a. A product of invertible n × n matrices is invertible, andthe inverse of the product is the product of their inverses in the same
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: A product of invertible $n \times n$ matrices is invertible, andthe inverse of the product is the product of their inverses in the same order.
The Statement is False. A product of invertible $n \times n$ matrices is invertible, and the inverse of the product is the product of their inverses in the reverse order.
Therefore
FALSE
Step 2: Part (b)
Statement: If A is invertible, then the inverse of A^{-1} is A itself.
The Statement is True according to theorem.
Therefore
TRUE
Step 3: Part (c)
Statement: If $A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$ and $ad = bc $; then A is not invertible.
The Statement is True. The necessary condition for an invertible matrix is that $ad - bc \ne 0$
Therefore
FALSE
Step 4: Part (d)
Statement: If A can be row reduced to the identity matrix, then A mustbe invertible.
The Statement is True by Inverse Matrix Theorem
Therefore
TRUE
Step 5: Part (e)
Statement: If A is invertible, then elementary row operations that reduce A to the identity $I_n$ also reduce $A^{-1}$ to $I_n$.
The Statement is False. By theorem 7 an matrix A $n \times n$ is invertible if and only if A is row equivalent to $I_n$, and in this case, any sequence of elementary row operations that reduces A to $I_n$ also transforms into $I_n$ into $A^{-1}$
Therefore
FALSE
we are given with some statements, that we have to find whether they are True or False
Step 1: Part (a)
Statement: A product of invertible $n \times n$ matrices is invertible, andthe inverse of the product is the product of their inverses in the same order.
The Statement is False. A product of invertible $n \times n$ matrices is invertible, and the inverse of the product is the product of their inverses in the reverse order.
Therefore
FALSE
Step 2: Part (b)
Statement: If A is invertible, then the inverse of A^{-1} is A itself.
The Statement is True according to theorem.
Therefore
TRUE
Step 3: Part (c)
Statement: If $A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$ and $ad = bc $; then A is not invertible.
The Statement is True. The necessary condition for an invertible matrix is that $ad - bc \ne 0$
Therefore
FALSE
Step 4: Part (d)
Statement: If A can be row reduced to the identity matrix, then A mustbe invertible.
The Statement is True by Inverse Matrix Theorem
Therefore
TRUE
Step 5: Part (e)
Statement: If A is invertible, then elementary row operations that reduce A to the identity $I_n$ also reduce $A^{-1}$ to $I_n$.
The Statement is False. By theorem 7 an matrix A $n \times n$ is invertible if and only if A is row equivalent to $I_n$, and in this case, any sequence of elementary row operations that reduces A to $I_n$ also transforms into $I_n$ into $A^{-1}$
Therefore
FALSE