Question

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

Answer 1

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