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Thread: 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 $\displaystyle A$, $\displaystyle C$ and $\displaystyle X$ as above:

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

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

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


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


    So, to answer your questions:

    1 True, because $\displaystyle AA^{-1} = A^{-1}A = I$, $\displaystyle II = I$ and $\displaystyle IX = X$.

    2 If you have to say True or False, then you'd say False. But in fact it's meaningless, because $\displaystyle BA^{-1}$ is impossible to form unless $\displaystyle n =1$ - the dimensions are incompatible. If you had to write a correct version then you'd say $\displaystyle AX = B \Rightarrow X = A^{-1}B$.

    3 The same applies here: neither $\displaystyle XC$ nor $\displaystyle DC^{-1}$ are possible products unless $\displaystyle n =1$. A correct statement would be $\displaystyle CX=D \Rightarrow X = C^{-1}D$, which is just the same as number 2 really.

    Grandad
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