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

1 Answer
Sep 13, 2015

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_(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 #(AB)_(mxxp)#

Furthermore, the transpose of a matrix is when rows become columns and columns become rows.
Hence, if #A_(mxxn)=>(A^T)_(nxxm)#

So in this particular question we have #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