Question

1 0 0 0 1 1 0 0 1. Let A be the 4 x 4 matrix: A= 1 1 1 0 1 1 1 1 (a) Find A-1 by hand. (b) Let T be defined by T(T) = Aē. Use your answer to (a) to find all vectors T E R4 such that: T(T) = 2 4 -9 0 (c) Which of the following statements are true? (Select all that apply) A. The columns of A form a basis of R4 B. The columns of A are linearly independent. C. The columns of A span R4

linear algebra

Answer 1

Given A= - a) find At we will nove reduce 9 To find At [Al I] to get [7 / At R Rg-Rig R3 R3-R, Ru Ry - R1 oo 0 (All] 01 OO 00 - - 1 0 (After row operation ) 6 0 0 1 - O - - RB - R₃-R2 Ry & Ry - R2 > 0 1 1 1 - 0 o 0 1 Ru Ry - R3 0 - O O 001- o O 0-1

ܙ ]- AM .. 0 0 6 Tx= Ax we have to find & EIRY such that Tx= AM - "ja !' ܕ 0 ܗ ܢ X 2 x = yܡ ܝܘܐ) 0 91 (AX)- 417 (ATA) X = Aly Id=Aly ܘ ܘ | | 9 * ) ܘ 6 ܘܘܐܗ ܕܐ 3 13

which of following statemente are teme e A.) The columns of a form a basis Ans. Tree, as rank of A = 4 also dim IRY = 4 column of a form basis a o fou IRY as 0 00 6) column of a are lineare independent. Any True, as AX=0 Ax = 0 x=0 At (AX) = Allo) as Aterest GX1 + 2X2 + (3X3 + Cuxy = 0 x=0 4 = 0,6-0, G=0, Cy=0 where xi: 8th column of x . The columns of A spanley Aus True, as rank of A= dim IR" =4. .