# Product of inverse matrices $(AB)^{-1}$

I am unsure how to go about doing this inverse product problem:

The question says to find the value of each matrix expression where A and B are the invertible 3 x 3 matrices such that $$A^{-1} = \left(\begin{array}{ccc}1& 2& 3\\ 2& 0& 1\\ 1& 1& -1\end{array}\right)$$ and $$B^{-1}=\left(\begin{array}{ccc}2 &-1 &3\\ 0& 0 &4\\ 3& -2 & 1\end{array}\right)$$

The actual question is to find $(AB)^{-1}$.

$(AB)^{-1}$ is just $A^{-1}B^{-1}$ and we already know matrices $A^{-1}$ and $B^{-1}$ so taking the product should give us the matrix $$\left(\begin{array}{ccc}11 &-7 &14\\ 7& -4 &7\\ -1& 1 & 6\end{array}\right)$$ yet the answer is $$\left(\begin{array}{ccc} 3 &7 &2 \\ 4& 4 &-4\\ 0 & 7 & 6 \end{array}\right)$$

What am I not understanding about the problem or what am I doing wrong? Isn't this just matrix multiplication?

Actually the inverse of matrix product does not work in that way. Suppose that we have two invertible matrices, $A$ and $B$. Then it holds: $$(AB)^{-1}=B^{-1}A^{-1},$$ and, in general: $$\left(\prod_{k=0}^NA_k\right)^{-1}=\prod_{k=0}^NA^{-1}_{N-k}$$

• For the sake of simplicity, let's assume $\prod_{k=0}^{N-1}A_i=A$ and $A_N=B$. You can easily verify that both A and B are invertible. Now you are looking for a matrix $C$ such that $C\cdot (AB) = I$. For the associative property lhs is equal to $(CA)B$. Since B is invertible, there exists a matrix $D$ such that $DB=I$. Since this matrix exists and is unique, it holds that $CA=D=B^{-1}$. Also, A is invertible, and we can multiply both lhs and rhs by $A^{-1}$ on the right. It finally holds: $C=B^{-1}A^{-1}$. By induction you prove the rest. Let me know if you need help with that – 7raiden7 Jan 25 '17 at 8:27
• Tomáš Zato: matrix multiplication is function composition. $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$ – kakashi10192020 Jun 14 '18 at 22:21

Note that the matrix multiplication is not commutative, i.e, you'll not always have: $AB = BA$.

Now, say the matrix $A$ has the inverse $A^{-1}$ (i.e $A.A^{-1} = A^{-1}.A = I$); and $B^{-1}$ is the inverse of $B$ (i.e $B.B^{-1} = B^{-1}.B = I$).

## Claim

$B^{-1}A^{-1}$ is the inverse of $AB$. So basically, what I need to prove is: $(B^{-1}A^{-1})(AB) = (AB)(B^{-1}A^{-1}) = I$.

Note that, although matrix multiplication is not commutative, it is however, associative. So:

• $(B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B = B^{-1}IB = (B^{-1}I)B = B^{-1}B=I$

• $(AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = A^{-1}IA = (A^{-1}I)A = A^{-1}A=I$

So, the inverse if $AB$ is indeed $B^{-1}A^{-1}$, and NOT $A^{-1}B^{-1}$.

• This is better than the accepted answer because it actually explains the WHY. – JP McCarthy Feb 11 '15 at 13:44

Not really. Matrices do not follow exponential laws. In fact, $(AB)^{-1}=B^{-1}A^{-1}$. Here is the proof:

Let $I$ be a 3 by 3 identity matrix. If $A$ and $B$ are 3 by 3 invertible matrices, then: \begin{align*} (AB)(AB)^{-1}&=I\\ (A^{-1}AB)(AB)^{-1}&=A^{-1}I\\ (IB)(AB)^{-1}&=A^{-1}\\ B(AB)^{-1}&=A^{-1}\\ B^{-1}B(AB)^{-1}&=B^{-1}A^{-1}\\ I(AB)^{-1}&=B^{-1}A^{-1}\\ (AB)^{-1}&=B^{-1}A^{-1} \end{align*}

$(AB)^{-1}$ is not equal to $A^{-1}B^{-1}$, but it is equal to $B^{-1}A^{-1}$.

• @user131127 If the answer is what you need, accept it and vote it up so the others know this is matter is closed. – 5xum Feb 24 '14 at 10:11

I have some personal opinions which might perfect it (many students made mistakes about this), in your case it works fine $$(AB)^{-1}=B^{-1}A^{-1}$$ Because you already have the fact that $A,B$ are both square matrix and invertible, but suppose $A$ is $m\times n$ matrix, and $B$ is $n\times m$ matrix, then $AB$ is $m\times m$ matrix which might be invertible, but in this case we don't even have square matrix so we could never have such fomular.