a. Can [ i ] be a column of a unitary matrix?
[-i ]
Justify your answer
b. Check whether or not the following linear system is consistent:
3+t = x+y+z, 2+2t= x+2y-z, 4-t= x-y +4z
1
Expert's answer
2021-07-22T11:39:58-0400
Part a
Yes, [i] can be a column at a unitary matrix as if we choose 1×1 matrix. I.e i=j, meaning that row = column.
So in this case we can say that [i] as a column of a unitary matrix.
Consqueantly
Yes, [-i] can be a column at a unitary matrix as if we choose 1×1 matrix. I.e -i=-j, meaning that row = column.
So in this case we can say that [-i] as a column of a unitary matrix.
Part b
x+y+z=3+tx+2y−z=2+2tx−y+4z=4−t
It can be written as
x+y+z−t=3x+2y−z−2t=2x−y+4z+t=4
Now we can convert it into matrix [A:b] form and do row reduction form
=⎣⎡11112−11−14−1∣3−2∣21∣4⎦⎤
=⎣⎡10011−21−23−1∣3−1∣−12∣1⎦⎤
=⎣⎡1000103−2−10∣4−1∣−10∣−1⎦⎤
=⎣⎡1000100010∣1−1∣10∣1⎦⎤
x= 1, y=1,z= 1
So the solution is consistent, the value of t is not fixed as t is a free variable. So it has infinitely many solutions.
Comments
Leave a comment