The matrix A is invertible with an eigenvector x corresponding to the eigenvalue λ. Which of the following is/are true? I : Ax=λx II : A −1
x= λ
1
​
x III : det(A−λI)=0 Matrix A is of size 3×3 and has eigenvalues λ 1
​
=1,λ 2
​
=0 and λ 3
​
=−1. Corresponding eigenvectors are v 1
​
= ⎝
⎛
​
1
1
1
​
⎠
⎞
​
,v 2
​
= ⎝
⎛
​
1
0
−1
​
⎠
⎞
​
and v 3
​
= ⎝
⎛
​
1
2
0
​
⎠
⎞
​
respectively. The vector x is such that x=v 1
​
+v 2
​
+v 3
​
Given that Ax=y, then the value of y T
is Select one: A. (−1,1,−1) B. (1,0,−1) c. (1,−1,1) D. (0,−1,1) E. (1,1,1) Consider the system written in augmented form as (A∣b). Using elementary row operations, the echelon system that is row-equivalent to (A∣b) is ⎝
⎛
​
1
0
0
​
−2
0
0
​
1
2
0
​
−1
3
0
​
0
−2
0
​
⎠
⎞
​
Which of the following is true? I : Rank(A)=2 II : The general solution has 2 free variables III : dim( Column Space )=2 The linear transformation T:R 3
→R 3
is such that T(x,y,z)=(0,0,z). What is kernel of T ? Select one: A. {(t,0,0)} where t∈R B. {(t,p,0)} where t,p∈R c. {(0,0,t)} where t∈R D. {(0,0,0)} E. {(t,t,0)} where t∈R