0

What 3 by 3 matrix E?

What 3 by 3 matrix E multiplies (x, y, z) to give (x, y, z+x)?

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
1
(x, y, z+x)
= (x, y, z) + (0, 0, x)
So, the matrix is
1 0 0....0 0 0....1 0 0
0 1 0 + 0 0 0 = 0 1 0
0 0 1....1 0 0....1 0 1
Kirill Z. | Physics, math tutor with great knowledge and teaching skillsPhysics, math tutor with great knowledge...
4.9 4.9 (174 lesson ratings) (174)
0
Basically, you have equation (x,y,z+x)=E(x,y,z) or (x,y,z)+(0,0,x)=E(x,y,z), which transforms into

(0,0,x)=(E-I)(x,y,z), where I is the unit matrix, I={(1,0,0);(0,1,0);(0,0,1)} Now denote E-I as U, then we have:

U11x+U12y+U13z=0
U21x+U22y+U23z=0
U31x+U32y+U33z=x

Since x, y, and x are arbitrary, the only way to satisfy those three equations is to set all Uij equal to zero, except U31, which shall be set equal to 1. So U={(0,0,0};(0,0,0);(1,0,0)}

Then E can be easily found by adding U and I, E=U+I
Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
5.0 5.0 (3 lesson ratings) (3)
0
E must be the identity matrix with an added 1 in the 1st column, 3rd row:

E = [[1,0,0],[0,1,0][1,0,1]]