Let A be a 5 by 7 , B be a 7 by 6 and C be a 6 by 5 matrix. How to determine the size of the following matrices ? AB, BA, A^TB, BC, ABC , CA ,B^TA , BC^T

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Let A be a 5 by 7 , B be a 7 by 6 and C be a 6 by 5 matrix. How to determine the size of the following matrices ? AB, BA, A^TB, BC, ABC , CA ,B^TA , BC^T

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Answer 1 · 2015-09-13T20:33:30

According to the theory of matrix multiplication, the matrix multiplication is only defined if B has the same number of columns as rows in A, that is, if $A_{m \times n} and B_{n \times p}$ $A_(mxxn) and B_(nxxp)$ , ie A has n columns and B has n rows, otherwise AB will not be defined.
If it is defined, as above, then the matrix AB will have m rows and p columns, ie ${(A B)}_{m \times p}$ $(AB)_(mxxp)$

Furthermore, the transpose of a matrix is when rows become columns and columns become rows.
Hence, if $A_{m \times n} \Rightarrow {(A^{T})}_{n \times m}^{T}$ $A_(mxxn)=>(A^T)_(nxxm)$

So in this particular question we have $A_{5 \times 7}, B_{7 \times 6}, C_{6 \times 5}$ $A_(5xx7), B_(7xx6), C_(6xx5)$

$therefore(AB)_(5xx6)$

$BA$ undefined.
$A^TB$ undefined

$BC_(7xx5)$

Since matrix multiplication is assosciative, $ABC = (AB)C = A(BC) therefore(ABC)_(5xx5)$

$(CA)_(6xx7)$

$B^TA$ is undefined

$BC^T$ is undefined

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Let A be a 5 by 7 , B be a 7 by 6 and C be a 6 by 5 matrix. How to determine the size of the following matrices ? AB, BA, A^TB, BC, ABC , CA ,B^TA , BC^T

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