Let matrix A 2 1 1 1 Find all matrices of the form M a b

\"Let matrix A= [ 2 1

1 1 ] Find all matrices of the form:

M=   [ a b

c d ] such that AM=MA\"

Please show all steps involved... Note both matrices are 2x2.

Solution

For AM to make sense, M has to be 2 x n matrix for some n. For MA to make sense, M has to be an m x 2 matrix. Thus M must be a 2x2 matrix. Thus, we may assume that M is the matrix:

[ a b ]
[ c d ]

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

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

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

For AM = MA, 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

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).

[ 0 1 -1 0 ]
[ 1 0 0 -1 ]
[ 1 0 -1 -1 ]
[ 0 1 -1 0 ]

Now, we perform gauss-jorden elimination to obtain the solution set:

[ 1 0 0 -1 ]
[ 1 0 -1 -1 ]
[ 0 1 -1 0 ]
[ 0 0 0 0 ]

[ 1 0 0 -1 ]
[ 0 0 -1 0 ]
[ 0 1 -1 0 ]
[ 0 0 0 0 ]

[ 1 0 0 -1 ]
[ 0 1 -1 0 ]
[ 0 0 1 0 ]
[ 0 0 0 0 ]

[ 1 0 0 -1 ]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 0 ]

This gives us:

a - d = 0
b = 0
c = 0

If we let d = t, we get:

a = t
b = 0
c = 0
d = t

Thus, our matrix M will be of the form:

[ t 0 ]
[ 0 t ]

That is, it will be any multiple of the identity matrix.