# 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#

a)

b)

d)

##### 1 Answer

Mar 18, 2017

**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