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 2.3 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 11E
Chapter:
Problem:
In Exercises 11 and 12, the matrices are all...
Step-by-Step Solution
Step 1
We are given with a matrix of size $n \times n$. We have to prove whether the given statements are True or False.
Step 2: (a)
If the equation $A{\bf{x}} = 0$ has only the trivial solution, then A is row equivalent to the $n \times n$ identity matrix.
From statements (b) and (d) of Theorem-8 we can say that if (b) is True then (d) is also True.
So from (d), if the equation $A{\bf{x}} = 0$ has only the trivial solution, then from (b), A is row equivalent to the $n \times n$ identity matrix.
TRUE
Step 3: (b)
If the columns of A span $R^n$, then the columns are linearly independent
From statements of Theorem-8 we can say that if statement (e) is True then (h) is also True.
So from (h), If the columns of A span $R^n$, then from (e) the columns of A form a linearly independent set.
TRUE
Step 4: (c)
If A is an $n \times n$ matrix, then the equation $A{\bf{x}} = b$ has at least one solution for each b in $R^n$.
For the given statement to be true, the matrix A has to be invertible. As this is not mentioned or inferred. So the statement is not True always.
FALSE
Step 5: (d)
If the equation $A{\bf{x}} = 0$ has a nontrivial solution, then A has fewer than n pivot positions
If the equation $A{\bf{x}} = 0$ has a non trivial solution, that means number of pivot rows are less then n. That means n has fewer then n pivot positions.
TRUE
Step 6: (e)
If $A^T$ is not invertible, then A is not invertible
From statements of Theorem-8 we can say that if statement (l) is False then (a) is also False.
So from (l), If $A^T$ is not invertible, then from (a) the matrix A also not invertible.
TRUE
ANSWER
(a) TRUE
(b) TRUE
(c) FALSE
(d) TRUE
(e) TRUE
We are given with a matrix of size $n \times n$. We have to prove whether the given statements are True or False.
Step 2: (a)
If the equation $A{\bf{x}} = 0$ has only the trivial solution, then A is row equivalent to the $n \times n$ identity matrix.
From statements (b) and (d) of Theorem-8 we can say that if (b) is True then (d) is also True.
So from (d), if the equation $A{\bf{x}} = 0$ has only the trivial solution, then from (b), A is row equivalent to the $n \times n$ identity matrix.
TRUE
Step 3: (b)
If the columns of A span $R^n$, then the columns are linearly independent
From statements of Theorem-8 we can say that if statement (e) is True then (h) is also True.
So from (h), If the columns of A span $R^n$, then from (e) the columns of A form a linearly independent set.
TRUE
Step 4: (c)
If A is an $n \times n$ matrix, then the equation $A{\bf{x}} = b$ has at least one solution for each b in $R^n$.
For the given statement to be true, the matrix A has to be invertible. As this is not mentioned or inferred. So the statement is not True always.
FALSE
Step 5: (d)
If the equation $A{\bf{x}} = 0$ has a nontrivial solution, then A has fewer than n pivot positions
If the equation $A{\bf{x}} = 0$ has a non trivial solution, that means number of pivot rows are less then n. That means n has fewer then n pivot positions.
TRUE
Step 6: (e)
If $A^T$ is not invertible, then A is not invertible
From statements of Theorem-8 we can say that if statement (l) is False then (a) is also False.
So from (l), If $A^T$ is not invertible, then from (a) the matrix A also not invertible.
TRUE
ANSWER
(a) TRUE
(b) TRUE
(c) FALSE
(d) TRUE
(e) TRUE