Question

please provide detailed and clear solutions for the following

2-6 3 2- 0 -103-5 Calculate the determinants of A and B -1 4 (use either appropriate row and coumn expansions or elementary row operations and the properties of determinants). Are A and B invertible? Calculate their inverses if they exist 1b. Are the columns of A linearly dependent or linearly independent? Find the dimension of Nul A and the rank of A. What can you say about the number of solutions of Ax b with unknown x E R and b R3 le. Do the columns of B span R' Find the dimension of Col B and the rank of B. What can you say about the number of solutions of By = c with unknown y E R' and c E R"? ld. Caleulate the determinants of A--1A, AT. B2 and AB or explain why they are undefined. le. Compute the volume of the parallelepiped P in R3 determined by the columns of A and of S(E) R is the matrix where E is the rectangular box 2-1 0,1 4, S:R transformation S(x)Ax, x ER3 and S(E) CR is the image of E under S

Answer 1

A)(o+y) 6(o-)+3(0-0) : 2(41+6 (-1) 40 = 8-622 103- 240 Awe have t s 26 an аз, аз 433 as 2 ⑥ (olumns 역, Noco, we know -hat matur A s )

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2- 1o o -7S ooOo Now, this in aduced Eshe oom Recal that a leading one te t nom-e enthy ng leade aethe pivot Col uumn spatejuut Lse the pivot column bom ogatm thebais column Spau an 2. oni Ltinitely many so

2. l6 6 3 -1 2-1 o 27 2 o o-2.6o i mot possibe ta to m multipu'atim mut equals o m2 So, A u not possible 3y3
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