Question

If `A,B,C` are matrices then determine if the following statements are correct?

a) `AB = BA`
b) `A(B^T-C^T) = AB^T-AC^T`
d) `(C^TB^T)A^T = B^TC^TA^T`

Answer 1

Part (a)
In general matrix multiplication is not commutative; ie

`AB != BA`

(a) is FALSE

Part (b)
Using properties of the transpose; we have:

`(A + B)^T = A^T + B^T`
`:. A(B-C)^T = A(B^T-C^T)`

Matrix addition is distributive;

`:. A(B^T-C^T) = AB^T-AC^T`

(b) is TRUE

Part (c)

Part (d)
Using properties of the transpose; we have:

`(AB)^T = B^TA^T`

`:. (ABC)^T = (A(BC))^T`
`" " = (BC)^TA^T`
`" " = (C^TB^T)A^T`

And matrix multiplication is associative but it is not commutative

`:. (C^TB^T)A^T = C^TB^TA^T != B^TC^TA^T`

(d) is FALSE