Consider the following matrices. Calculate the results of the operations below, where the operations are allowed.

Question

I have added a image of the question below, but to explain: suppose you have two matrices, one called C and another called B, how do you work out (CB)^T? is it a matter of multiplying C and B and then transposing or is it a matter of B transposed multiplied by C transposed since order of multiplication matters a lot as well as row/columns!

I think I have i) and ii) understood however it is the questions that follow after, especially the fourth

---

My question is to do with order, AB^T = B^T * A^T but how does this work? Are the transposed version worked out first before multiplying or can one simply do B*A and then find transposed version?

Question

Answer 1

"$(CB)^T$" means "multiply $B$ by $C$ on the left, then take the transpose of the result". This is what parentheses mean: do what is inside the parentheses before what is on the outside.

That said, mathematical operations often have properties that tell you two different calculations will give the same result. For instance:

• Associative laws: $(a + b) + c = a + (b + c)$. It does not matter if you add $a$ and $b$ first, then add $c$, or if you add $b$ and $c$ first, then add $a$. You get the same answer with either calculation. $(1 + 2) + 3 = (3) + 3 = 6$ and $1 + (2 + 3) = 1 + 5 = 6$. Two different calculations, but the final result for either is the same.
• Distributive law: $a(b + c) = ab + ac$. $2(3 + 4) = 2(7) = 14$ and $2\cdot 3 + 2\cdot 4 = 6 + 8 = 14$. Once again, these are two different calculations, but because the distributive law holds, we know it doesn't matter which calculation we actually do, the final result will be the same.

$(AB)^T = B^TA^T$ is just another example of such a law: The left side tells you to multiply $AB$ first, then transpose. The right side says to transpose $A$ and $B$ first, and then multiply them together in the reverse order. The law tells you it doesn't matter which calculation you do, the final result will be the same either way.

This law isn't just something someone made up on a whim (and neither are the other laws). It is a consequence of how matrix multiplication and transposition are defined, as is demonstrated in Nightflight's linked thread.