Question

Please help me out ASAP with this statement about Matrix?

Does matrix multiplication yield a Matrix or a Vector or a Scalar number?

See for instance if two matrices (Column vectors) are given matrix `A=((5),(0),(0))` and Matrix `B= ((0),(6),(8))` and I want to take the dot product `vecA.vecB` ; then it is defined as the Matrix product `B^TA` so it comes out to be a Scalar real number ......................

BUT actually it is Matrix multiplication so the answer should have been a Matrix.!!

Please explain me where I am wrong or what is the correct explanation for this ?

Answer 1

Technically speaking, your `B^TA` is a `1times 1` matrix - but there is a natural 1-1 correspondence between `1 times 1` real matrices and real numbers : `(a) mapsto a` - that helps us identify such matrices with numbers. So, you can think of the result as either a `1 times 1` matrix or a number - the choice is yours!

Answer 2

Matrix multiplication, `AB`, requires that matrices `A` and `B` be of dimensions `m xx n` and an `n xx p`; the result is always a matrix of dimension `m xx p`.

Explanation:

Extending the above fundamental principle, we conclude that `A = ((5),(0),(0))` and `B = ((0),(6),(8))` are column vectors, not matrices , because we can perform the dot product which always yields a scalar. Matrix multiplication always yields a matrix.

If we had a matrix `C` of dimension `mxx3`, then we could treat `A` and `B` as `3xx1` matrices and we could multiply `CA` or `CB` and obtain an `mxx1` matrix.