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A simple question

is a (5x2)*(2x5) = a (5x5) matrix?

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    Yes. ${}{}{}{}{}$2011-05-02
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    http://en.wikipedia.org/wiki/Matrix_multiplication#Technical_Details2011-05-02
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    Yes, the matrix product $AB$ is defined when $A$ has dimensions $n\times m$ and $B$ has dimensions $m\times k$ for any integers $n,m,k$, and the resulting matrix will have dimensions $n\times k$.2011-05-02
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    How about a (3X2) * (5X3) is this a 3x5?... this is my last question to finally understand this2011-05-02
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    @edprof: that product is not defined in that order (the dimensions $2$ and $5$ on the inside don't match). In the other order, it's $5 \times 2$.2011-05-02
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    @edprof: As Qiachu and matt have said, the right-most dimension of the matrix on the left must match the left-most dimension of the matrix on the right, or else the product is meaningless. The dimension on the left represents the number of rows in a matrix; the dimension on the right represents the number of columns of the matrix. Check out the link provided by DJC for more details.2011-05-02
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    @All: why did 'community' bring this back to the front page?2011-06-04
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    @mixedmath, my understanding is that "Community" somehow selects questions where no answer has been accepted and resurrects them in the hope that the questioner will find some answer, old or new, acceptable.2011-06-04

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A matrix of order $m \times n$; where $m$ is the number of rows and $n$ the number of columns 'represents' a linear map that takes input from an $n-$ dimensional vector space and outputs in a $m-$ dimensional vector space. The rules of multiplying matrices comes from the fact that we want matrix multiplication to reflect the composition of two maps. Since, for composition of maps to work, the (space) for the output of the first map should coincide with the (space) of the input of the next map. Hence the rule that the column of one matrix should coincide with the row of the other.

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Yes, It is indeed a $5 \times 5$ matrix. For any matrix multiplication to be defined for some arbitrary matrices $A$ and $B$, $A$ could be a $m \times n$ and $B$ could be a $p \times q$, in this order, for the matrix multiplication to be defined. The amount of columns of matrix $A$ must equal the amount of rows of matrix $B$, (i.e., $n=p$).

So a $~\Rightarrow~$ $m \times n$ $\cdot$ $p \times q$ will result in a $m \times q$ matrix $\iff~~n=p$

$\underline{\text{Example:}}~~$ $3 \times 4$ $\cdot$ $4 \times 3$ = $3 \times 3$ matrix.

I hope that this helps out a bit.

Good~Luck.

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    I'm sure you know this, but you don't need the number of rows in $A$ and the number of columns in $B$ to both be $m$...2011-05-02
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    Yes, the answer as written is slightly misleading (personally, reading it, I wondered during some time if the product considered was BA...). I would suggest rephrasing.2011-05-02
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    @Matt: Yes, indeed you are correct. I just arbitrarily came up with those notations off the top of my head. I can edit it if it is misleading to any.2011-05-05
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Yes.. the way that i learnt it, when a 5*2 matrix is mutiplied into a 2*5 matrix, the middle number get taken away, leaving only the outer numbers, so in this case, the twos get taken away, leaving only the fives..