A matrix multiplied by a vector will result in a vector (through the vector may actually be a 1 x 1 vector, which is a number). In general an n x m matrix times an m vector will result in an n vector. Each entry of the resulting vector is found by multiplying the entries of each row of the mattrix by the entries of the vector and adding them.
Example: The 2 x 3 matrix
is to be multiplied by the 3 vector
The result will be a 2 vector.
The first entry in the resulting vector will be found by multiplying the entries in the first row by the entries in the vector and adding them.
We obtain 2*0+5*1+4*-2 = -3.
The second entry in the resulting vector will be found by multiplying the entries in the second row by the entries in the vector and adding them.
We obtain 1*0+0*1+-3*-2 = 6.
The result of the multiplication is