Find V1 V2 and V1 V2 Let A and B be two n x n matrices sati

Find V_1 V_2 and V_1 + V_2. Let A and B be two n x n matrices satisfying AB = BA. Show that Nu\\(A) + Nul(B) Nul(AB). Construct the following examples with the required properties.

Solution

solution-: For AB to make sense, B has to be n x n matrix for some n. For BA to make sense, B has to be an n x 2 matrix. Thus B must be a 2x2 matrix. Thus, we may assume that B is the matrix:
[ a b ]
[ c d ]

for some real a, b, c, d. We have:

AB = [ 2 1 ][ a b ] = [ 2a+c b+d ]
[ 1 1 ][ c d ] .. [ a+c b+d ]

BA = [ a b ][ 2 1 ] = [ 2a+b a+b ]
[ c d ][ 1 1 ] .. [ 2c+d c+d ]

For AB = BA, we need every element to be equal. Thus, we can expand this into a system of four simultaneous equations:

2a + c = 2a + b
b + d = a + b
a + c = 2c + d
b + d = c + d

so we seethat If we take everything to one side, we get a system of four linear equations:

b - c = 0 ... (1)
a - d = 0 ... (2)
a - c - d = 0 ... (3)
b - c = 0 ... (4)

As you can see, (1) = (3), so this system is dependent, and will have infinitely many solutions. Let\'s transform this homogeneous system into a matrix. We don\'t need an augmented matrix, since the system is homogeneous (the 0s add no information).

Hence NulA+ nulB= Nul AB