Thread: 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.

Thanks in advance.

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.

Thanks in advance.

First a few basics. With , and as above:

* If has an inverse (and it might not!), then , where is the identity matrix.

* The identity matrix doesn't change any matrix that it multiplies. So, for instance, and .

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

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

So, to answer your questions:

1 True, because , and .

2 If you have to say True or False, then you'd say False. But in fact it's meaningless, because is impossible to form unless - the dimensions are incompatible. If you had to write a correct version then you'd say .

3 The same applies here: neither nor are possible products unless . A correct statement would be , which is just the same as number 2 really.

Grandad