invertible matrix, prove that 5A is an invertible matrix. ystem stem = [³] 4x = b₁, the same ix is the art (a) by 3 b4]. e and D In Exercises 9 and 10, mark each statement True or False. Justify each answer. 9. a In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true. b. If A and B are n x n and invertible, then A¹B¹ is the inverse of AB. c. If A = = [a b] C e. 10. /a. d. If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in R". Each elementary matrix is invertible. w A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order. b. If A is invertible, then the inverse of A¹ is A itself. c. If A = a [2 C and abcd # 0, then A is invertible. T b and ad bc, then A is not invertible. 279 d. If A can be row reduced to the identity matrix, then A must be invertible. e. If A is invertible, then elementary row operations that reduce A to the identity In also reduce A¹ to In. 11. Let A be an invertible n x n matrix, and let B be an n x p matrix. Show that the equation AX = B has a unique solu- tion A-¹ B. 12. Let A be an invertible n x n matrix, and let B be an n x p ma- trix. Explain why A¹B can be computed by row reduction:
invertible matrix, prove that 5A is an invertible matrix. ystem stem = [³] 4x = b₁, the same ix is the art (a) by 3 b4]. e and D In Exercises 9 and 10, mark each statement True or False. Justify each answer. 9. a In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true. b. If A and B are n x n and invertible, then A¹B¹ is the inverse of AB. c. If A = = [a b] C e. 10. /a. d. If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in R". Each elementary matrix is invertible. w A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order. b. If A is invertible, then the inverse of A¹ is A itself. c. If A = a [2 C and abcd # 0, then A is invertible. T b and ad bc, then A is not invertible. 279 d. If A can be row reduced to the identity matrix, then A must be invertible. e. If A is invertible, then elementary row operations that reduce A to the identity In also reduce A¹ to In. 11. Let A be an invertible n x n matrix, and let B be an n x p matrix. Show that the equation AX = B has a unique solu- tion A-¹ B. 12. Let A be an invertible n x n matrix, and let B be an n x p ma- trix. Explain why A¹B can be computed by row reduction:
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.2: Properties Of Matrrix Operations
Problem 69E: GuidedProof Prove that if A is an mn matrix, then AAT and ATA are symmetric matrices. Getting...
Related questions
Question
9 (a,b,c,d,e)
![invertible matrix, prove that 5A is an invertible matrix.
ystem
stem
= [³].
4x = b₁,
the same
ix is the
art (a) by
3 b4].
e and D
In Exercises 9 and 10, mark each statement True or False. Justify
each answer.
9. a In order for a matrix B to be the inverse of A, both
equations AB = I and BA = I must be true.
b. If A and B are n x n and invertible, then A¹B¹ is the
inverse of AB.
bm
c. If A =
e.
10. /a.
b
[a
[] and ab
d
d. If A is an invertible nxn matrix, then the equation
Ax=b is consistent for each b in R".
Each elementary matrix is invertible.
A product of invertible n x n matrices is invertible, and
the inverse of the product is the product of their inverses
in the same order.
b. If A is invertible, then the inverse of A¹ is A itself.
c. If A =
a
[2
C
and abcd # 0, then A is invertible.
worshimi
21 A
b
Win
and ad bc, then A is not invertible.
279
d. If A can be row reduced to the identity matrix, then A must
be invertible.
e. If A is invertible, then elementary row operations that
reduce A to the identity I, also reduce A¹ to In.
11. Let A be an invertible n x n matrix, and let B be an n x p
matrix. Show that the equation AX = B has a unique solu-
tion A-¹ B.
12. Let A be an invertible n x n matrix, and let B be an n x p ma-
trix. Explain why A¹B can be computed by row reduction:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05db57b3-41b7-4dd1-8647-35c67a680918%2F5bf8f557-a54b-468a-9c21-be13da21ab6c%2Fj478a2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:invertible matrix, prove that 5A is an invertible matrix.
ystem
stem
= [³].
4x = b₁,
the same
ix is the
art (a) by
3 b4].
e and D
In Exercises 9 and 10, mark each statement True or False. Justify
each answer.
9. a In order for a matrix B to be the inverse of A, both
equations AB = I and BA = I must be true.
b. If A and B are n x n and invertible, then A¹B¹ is the
inverse of AB.
bm
c. If A =
e.
10. /a.
b
[a
[] and ab
d
d. If A is an invertible nxn matrix, then the equation
Ax=b is consistent for each b in R".
Each elementary matrix is invertible.
A product of invertible n x n matrices is invertible, and
the inverse of the product is the product of their inverses
in the same order.
b. If A is invertible, then the inverse of A¹ is A itself.
c. If A =
a
[2
C
and abcd # 0, then A is invertible.
worshimi
21 A
b
Win
and ad bc, then A is not invertible.
279
d. If A can be row reduced to the identity matrix, then A must
be invertible.
e. If A is invertible, then elementary row operations that
reduce A to the identity I, also reduce A¹ to In.
11. Let A be an invertible n x n matrix, and let B be an n x p
matrix. Show that the equation AX = B has a unique solu-
tion A-¹ B.
12. Let A be an invertible n x n matrix, and let B be an n x p ma-
trix. Explain why A¹B can be computed by row reduction:
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