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

Question

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

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Answer 1 · 2017-03-18T02:20:08

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

$A B \neq B A$ $AB != BA$

(a) is FALSE

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

${(A + B)}^{T} = A^{T} + B^{T}$ $(A + B)^T = A^T + B^T$
$:. A(B-C)^T = A(B^T-C^T)$

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