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

## a) $A B = B A$ b) $A \left({B}^{T} - {C}^{T}\right) = A {B}^{T} - A {C}^{T}$ d) $\left({C}^{T} {B}^{T}\right) {A}^{T} = {B}^{T} {C}^{T} {A}^{T}$

Mar 18, 2017

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

$A B \ne B A$

(a) is FALSE

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

${\left(A + B\right)}^{T} = {A}^{T} + {B}^{T}$
$\therefore A {\left(B - C\right)}^{T} = A \left({B}^{T} - {C}^{T}\right)$

$\therefore A \left({B}^{T} - {C}^{T}\right) = A {B}^{T} - A {C}^{T}$

(b) is TRUE

Part (c)

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

${\left(A B\right)}^{T} = {B}^{T} {A}^{T}$

$\therefore {\left(A B C\right)}^{T} = {\left(A \left(B C\right)\right)}^{T}$
$\text{ } = {\left(B C\right)}^{T} {A}^{T}$
$\text{ } = \left({C}^{T} {B}^{T}\right) {A}^{T}$

And matrix multiplication is associative but it is not commutative

$\therefore \left({C}^{T} {B}^{T}\right) {A}^{T} = {C}^{T} {B}^{T} {A}^{T} \ne {B}^{T} {C}^{T} {A}^{T}$

(d) is FALSE