# Thread: Matrices and Inverses - Generalizing?

1. ## Matrices and Inverses - Generalizing?

Hello everyone...I'm studying for a midterm exam and I'm having a lot of trouble understanding this particular excercise:

Suppose matrices A and C have size n x n and matrices X, B and D have size n x 1.

1) True or False: AA -¹ A -¹ AX = X
2) True or False: If AX = B, then X = BA -¹
3) True or False: If XC = D, then X = DC -¹

I honestly do not even know where to begin with this...I'd really appreciate help figuring this problem out.

2. ## Matrices and Inverses

Hello luxdelux
Originally Posted by luxdelux
Hello everyone...I'm studying for a midterm exam and I'm having a lot of trouble understanding this particular excercise:

Suppose matrices A and C have size n x n and matrices X, B and D have size n x 1.

1) True or False: AA -¹ A -¹ AX = X
2) True or False: If AX = B, then X = BA -¹
3) True or False: If XC = D, then X = DC -¹

I honestly do not even know where to begin with this...I'd really appreciate help figuring this problem out.

First a few basics. With $A$, $C$ and $X$ as above:

* If $A$ has an inverse (and it might not!), then $AA^{-1} = A^{-1}A = I$, where $I$ is the $n \times n$ identity matrix.

* The identity matrix doesn't change any matrix that it multiplies. So, for instance, $IA = A$ and $IX = X$.

* Matrix multiplication is not commutative; that is to say, the matrix product $AC$ is not in general equal to the product $CA$.

* When multiplying two matrices, $L$ and $M$, say, the matrices must have 'compatible' dimensions. That is, in order to form the product $LM$, the number of columns in $L$ must be the same as the number of rows in $M$. So if $L$ is a $p \times q$ matrix, $M$ must be a $q \times r$ matrix, for some values of $p$, $q$ and $r$.

1 True, because $AA^{-1} = A^{-1}A = I$, $II = I$ and $IX = X$.
2 If you have to say True or False, then you'd say False. But in fact it's meaningless, because $BA^{-1}$ is impossible to form unless $n =1$ - the dimensions are incompatible. If you had to write a correct version then you'd say $AX = B \Rightarrow X = A^{-1}B$.
3 The same applies here: neither $XC$ nor $DC^{-1}$ are possible products unless $n =1$. A correct statement would be $CX=D \Rightarrow X = C^{-1}D$, which is just the same as number 2 really.