Linear algebra: please solve q35,37 and 40 correctly. Theorem is also attached. THEOREM 17ProofLet A and B be (n x n) matrices, each of which has an inverse. Then:A¹ has an inverse, and (A-1)-¹ = A.1-11.2.AB has an inverse, and (AB)-¹ = B-¹A-¹.3.If k is a nonzero scalar, then kA has an inverse, and (kA)−¹ = (1/k)A¯¹.4. AT has an inverse, and (AT)-¹ = (A−¹)ª.1. Since AA-¹ = A-¹A = I, the inverse of A-¹ is A; that is, (A¯¹)−¹ = A.2. Note that (AB)(B−¹A−¹) = A(BB¯¹)A¯¹ = A(IA¯¹) = AA-¹ = I. Simi-larly, (B-¹A−¹)(AB) = I, so, by Definition 13, B¯¹A-¹ is the inverse for AB.Thus (AB)-¹ = B-¹A-¹.3. The proof of property 3 is similar to the proofs given for properties 1 and 2 andis left as an exercise. The following matrices are used in Exercises 35-45.3112- [8₂] - - [1 ]·~~- - [ + ][³B-1 1202(9)In Exercises 35-45, use Theorem 17 and the matrices in(9) to form Q-¹, where Q is the given matrix.35. Q = AC36. Q = CA37. Q = AT38. Q = ATC39. Q = CTAT41. Q = CB-143. Q = 2A45. Q = (AC)B-¹40. Q = B-¹A42. Q = B-144. Q = 10C

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear algebra: please solve q35,37 and 40 correctly. Theorem is also attached.

THEOREM 17
Proof
Let A and B be (n x n) matrices, each of which has an inverse. Then:
A¹ has an inverse, and (A-1)-¹ = A.
1-1
1.
2.
AB has an inverse, and (AB)-¹ = B-¹A-¹.
3.
If k is a nonzero scalar, then kA has an inverse, and (kA)−¹ = (1/k)A¯¹.
4. AT has an inverse, and (AT)-¹ = (A−¹)ª.
1. Since AA-¹ = A-¹A = I, the inverse of A-¹ is A; that is, (A¯¹)−¹ = A.
2. Note that (AB)(B−¹A−¹) = A(BB¯¹)A¯¹ = A(IA¯¹) = AA-¹ = I. Simi-
larly, (B-¹A−¹)(AB) = I, so, by Definition 13, B¯¹A-¹ is the inverse for AB.
Thus (AB)-¹ = B-¹A-¹.
3. The proof of property 3 is similar to the proofs given for properties 1 and 2 and
is left as an exercise.

The following matrices are used in Exercises 35-45.
31
12
- [8₂] - - [1 ]·~~- - [ + ]
[³
B
-1 1
2
02
(9)
In Exercises 35-45, use Theorem 17 and the matrices in
(9) to form Q-¹, where Q is the given matrix.
35. Q = AC
36. Q = CA
37. Q = AT
38. Q = ATC
39. Q = CTAT
41. Q = CB-1
43. Q = 2A
45. Q = (AC)B-¹
40. Q = B-¹A
42. Q = B-1
44. Q = 10C

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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