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 ?

2 Answers
Ananda Dasgupta
Apr 26, 2018

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!

Douglas K.
Apr 26, 2018

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.

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