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Math Help - Matrices and Inverses - Generalizing?

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

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

    Thanks in advance.
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  2. #2
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    Matrices and Inverses

    Hello luxdelux
    Quote Originally Posted by luxdelux View Post
    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 - A - AX = X
    2) True or False: If AX = B, then X = BA -
    3) True or False: If XC = D, then X = DC -

    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 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.


    So, to answer your questions:

    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.

    Grandad
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