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1. If A and B are real square matrices, show that a) b) c) (A+AT) is symmetric, (A -AT) is skew-symmetric. AAT and AA are symmetric. A is symmetric if A is symmetric or skew-symmetric.

1. If A and B are real square matrices, show that a) b) c) (A+AT) is symmetric, (A -AT) is skew-symmetric. AAT and AA are symmetric. A is symmetric if A is symmetric or skew-symmetric.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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1. If A and B are real square matrices, show that 2. (A+AT) is symmetric, (A - AT) is skew-symmetric. AAT and AA are symmetric. A² is symmetric if A is symmetric or skew-symmetric. 1 2 If A = 3 4 2 1 -3 2 (A+B)¹ = A¹+ BT. (AB)¹ =BTAT. trace(AB)=trace(BA). c) 3. Expand (A+B)³ if a) A, B are commute - 1 7. Find the result of b) A, B are not commute 4. Show that (A − kI), (B+kI) commute for every scalar k if and only if A and B commute. 5. IF A and B are regular matrices, show that a) (AB)`¹ = B-¹ A-¹. b) (A-¹)¹ = (A¹)-¹. 10. For A = 6. A matrix A is called orthogonal if AA¹ = A¹A=I. If A is orthogonal, show that: a) |A| = ±1. b) A and A¹ are also orthogonal. 9. Find the determinant value : [2 1 01 1 10 32 0 2 1 1 0 0 1 2 3 12 9. Given A, B and C are 3×3 matrices, |4|=2, |2B¯¹|= 48 and A B C = I, Find|C| . 2 2 4 6 5 -2 4 3 2 -4 7 1 2 1 0 1 3 -1 -4 0 1 1 1 0 1 0 2 3 and B = 0 a) Adjoint method. b) Gauss Jordan Method. 2, verify that: 1 2 1 Find A¹ by using
Transcribed Image Text:1. If A and B are real square matrices, show that 2. (A+AT) is symmetric, (A - AT) is skew-symmetric. AAT and AA are symmetric. A² is symmetric if A is symmetric or skew-symmetric. 1 2 If A = 3 4 2 1 -3 2 (A+B)¹ = A¹+ BT. (AB)¹ =BTAT. trace(AB)=trace(BA). c) 3. Expand (A+B)³ if a) A, B are commute - 1 7. Find the result of b) A, B are not commute 4. Show that (A − kI), (B+kI) commute for every scalar k if and only if A and B commute. 5. IF A and B are regular matrices, show that a) (AB)`¹ = B-¹ A-¹. b) (A-¹)¹ = (A¹)-¹. 10. For A = 6. A matrix A is called orthogonal if AA¹ = A¹A=I. If A is orthogonal, show that: a) |A| = ±1. b) A and A¹ are also orthogonal. 9. Find the determinant value : [2 1 01 1 10 32 0 2 1 1 0 0 1 2 3 12 9. Given A, B and C are 3×3 matrices, |4|=2, |2B¯¹|= 48 and A B C = I, Find|C| . 2 2 4 6 5 -2 4 3 2 -4 7 1 2 1 0 1 3 -1 -4 0 1 1 1 0 1 0 2 3 and B = 0 a) Adjoint method. b) Gauss Jordan Method. 2, verify that: 1 2 1 Find A¹ by using
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