Question

[1 2 37 1. Is the matrix 1 0 1 invertible? If yes, what is its inverse? [O 2 -1 2. A matrix is called symmetric if At = A. What can you say about the shape of a symmetric matrix? Give an example of a symmetric matrix that is not a zero matrix. 3. A matrix is called anti-symmetric if A= -A. What can you say about the shape of an anti- symmetric matrix? Give an example of an anti-symmetric matrix that is not a zero matrix. 4. Show that AA is symmetric for every matrix A. 5. Find a 2 x 2 matrix A such that A² = I2, but A + +12. (Hint: you can do this algebraically, or geometrically.) For all the remaining questions, let n > 2 and let A and B ben xn matrices. 6. Does the equation A(B – In) + (In – B)A = On,n always hold? Either prove it or give a counter-example. 7. If A and B are invertible, does that imply that AB is invertible? What about the other way around? 8. Suppose that AB = A + B. Compute (In – A)(In – B). Use this to show that AB = BA.

Answer 1

as per Chegg guidelines I am solving first 4 questions ..please submit another question for remaining ..

A=5 l 237 lol 02 det (Al 160-2) -21-1)+312) = -2 +246 to So, matrix is invectible

r 1 2 3 1 Too R2ER2-R RIER3TR2 er la 3 loo 1 O -2 -2 - Lo oo -3 | AL R3E-Y30R] are 23 I loo7 1 0 -2 -2 | Lo

R2ER2 + R3 PER -3R3 rii ? Ol oli 7 1 0 -2 0 -3 Y2 -23 looi Y3 -43 -Y3 - R2E - 1 R2 0107616 Y3 Oo! Y -Y3 -43 -2.R2 riool -43 43 437 o lol Y6 -1/6 Y3 0 13 -43 -43 te rall 4/3 77 / Y6 Y6 Y3 L / -43 -13) ACA let a= r Example 103 frien I shape is always square

© Anti -syntyetiniai REA chef:23 shape is square An (5 %) [24] =65 -8 = (in ] [B=8 a HAT is symmetic.