# 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

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 ${A}_{m \times n} \mathmr{and} {B}_{n \times p}$, 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 ${\left(A B\right)}_{m \times p}$

Furthermore, the transpose of a matrix is when rows become columns and columns become rows.
Hence, if ${A}_{m \times n} \implies {\left({A}^{T}\right)}_{n \times m}$

So in this particular question we have ${A}_{5 \times 7} , {B}_{7 \times 6} , {C}_{6 \times 5}$

$\therefore {\left(A B\right)}_{5 \times 6}$

$B A$ undefined.
${A}^{T} B$ undefined

$B {C}_{7 \times 5}$

Since matrix multiplication is assosciative, $A B C = \left(A B\right) C = A \left(B C\right) \therefore {\left(A B C\right)}_{5 \times 5}$

${\left(C A\right)}_{6 \times 7}$

${B}^{T} A$ is undefined

$B {C}^{T}$ is undefined