Prove the following properties of matrices

Question

Let $A$ be a $M \times N$ matrix and $B$ a $N \times P$ matrix. Prove that $(AB)^T = B^TA^T$.
Use the result in Problem 1 and the associative property of matrix multiplication to show that $(ABC)^T$ = $C^TB^TA^T$

I already have a drafted anwswer in Problem 1. I let $C = AB$ and use the definition of matrix multiplication that the $(i,j)^{th}$ entry of $C^T$ must be equal to the $(i,j)^{th}$ entry of $B^TA^T$. Then I was having a problem with Problem 2.

Could you help me out?

(A(BC))'=(BC)'A'=C'B'A' – Angela Pretorius Jul 20 '13 at 06:25 — Angela Pretorius, Jul 20 '13 at 06:25

score 4 · Answer 1 · answered Jul 20 '13 at 06:25

4

HINT: Your approach to Problem 1 is sound, though writing down the details takes a little concentration. For Problem 2, let $X=AB$; then $(ABC)'=(XC)'$, and you can use Problem 1 a couple of times to finish the argument.

answered Jul 20 '13 at 06:25

Brian M. Scott

616,228

1

Or you could just let Angela do the whole problem for you --- then you wouldn't have to think at all. – Gerry Myerson Jul 20 '13 at 06:27

score 2 · Answer 2 · answered Jul 20 '13 at 06:26

Hint: Let $X=BC$ (this is just a change in notation). We need to prove that:

(1) $(AX)'=C'B'A'$.

You've already proven:

(2) $X'=C'B'$.

Therefore, (1) is equivalent to:

(3) (I'll let you rewrite (1) using (2))

And you've already proven (3)!

Exercise 1: If you're familiar with the principle of mathematical induction, then can you prove that $(A_1\cdots A_n)'=A_n'\cdots A_1'$ for all natural numbers $n$ using the idea I presented above?

I hope this helps!

Prove the following properties of matrices

2 Answers2