Question

Determine the dimensions of Nul A and Col A for the matrix shown below. 1 2 -4 5 -2 6 - 1 0 0 0 0 0 0 A= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The dimension of Nul A is 6, and the dimension of Col A is 0 Find an invertible matrix P and a matrix Cof the form 122-39 such that A= has the form A=PCP" 24 -38

need help calculating Nul A and dimension col A. those are answers I got .
question 2 need help with invertible matrix p and c from the form given below

Answer 1

Problem 1) The given matrix A has a nonzero row and 3 zero rows. The rank of A is 1.

Nul(A)= (number of columns of A)-rank(A) = 7-1 = 6

Col(A) is generated by the vector $\small \bigl(\begin{smallmatrix} 1\\ 0\\ 0\\ 0 \end{smallmatrix}\bigr)$ . Dimension of Col(A)=1

Problem 2)

2.4 A=122 -3.97 -3.8 Charectarinfie folynomical of A i det( A-x7) - 2-2-2 -3.9 2-4 10 10 10 -3.8-a -> (262-2)(-3.8-2)+(3.9x2:4)=0 => (x+3.8) (M-2-2) + 9:36 -0 7 mnt 1.62 - 8-36 + 9-36=B >"71063 +1=0 52 +82 +5-0 X = -8+ V64-100 --S + Eigen values are -0.8+ 0.6 -0.8 -0.67 Eigen vector corresponding to -0-870.67 is founded by Av=Xv or (A-1)v-o cohere va :( ] (W)=(8) -3.8-(-0.8 +0.60) 3-0061 3) -(0) to N2 2.2 - (-0.8+.60) -34 2.4 -3.9 [ ] ( 204 -3-0-67 >(3-0.61) v - 3.9 V2 = 0 2:41 +-3-06) V- -) VI V2 3.9 3-9(3+0.6;) 9.36 3-0.6i V + 3 2-4 Vz 2-4 :) - 14 ) V2 vi = 5 9 f + vector in Eigen 1 h ( The eigen vector wectar corresponding to (-0.8-0-6i) in the Conjugate of the above vector le. (**)

06 Choose the matin C to be -008 0.6 -0.8 Ly yo -- Then p = A= pcpY as required.

C = [-0.8 -0.6) 0.6 -0.8 and P= so that A= PCP