```Question 292657
{{{(matrix(1,4,
4, 1, 0, 2))*

(matrix(4,3,

1, 0, 1,
2, -1, 0,
3,  5, 1,
1,  3, 0)) }}}

This is a {{{red(1)}}}{{{x}}}{{{green(4)}}} matrix multiplied by a {{{green(4)}}}{{{x}}}{{{red(3)}}}.

The multiplication is defined because the two (green) inner
dimensions are both {{{green(4)}}} and the product matrix will
have the outer (red) dimensions {{{red(1)}}}{{{x}}}{{{red(3)}}}.

So we make a blank {{{1}}}{{{x}}}{{{3}}} matrix to fill in:

{{{(matrix(1,4,
4, 1, 0, 2))*

(matrix(4,3,

1, 0, 1,
2, -1, 0,
3,  5, 1,
1,  3, 0))

=

(matrix(1,3,"_","_","_"))

}}}

The first element to fill in is the 1st (only) row and 1st column.

So we multiply the first (only) row of the first matrix by the
first column of the second matrix and then add them like this:

{{{4*1 + 1*2 + 0*3 + 2*1 = 4 + 2 + 0 + 2 = 8}}}, so
we put {{{red(8)}}} in the first blank of the product matrix:

{{{(matrix(1,4,
4, 1, 0, 2))*

(matrix(4,3,

1, 0, 1,
2, -1, 0,
3,  5, 1,
1,  3, 0))

=

(matrix(1,3,red(8),"_","_"))

}}}

The second element to fill in is the 1st (only) row and 2nd column.

So we multiply the first (only) row of the first matrix by the
2nd column of the second matrix and then add them like this:

{{{4*0 + 1*(-1) + 0*5 + 2*3 = 0 - 1 + 0 + 6 = 5}}}, so
we put {{{red(5)}}} in the second blank of the product matrix:

{{{(matrix(1,4,
4, 1, 0, 2))*

(matrix(4,3,

1, 0, 1,
2, -1, 0,
3,  5, 1,
1,  3, 0))

=

(matrix(1,3,red(8),red(5),"_"))

}}}

The third element to fill in is the 1st (only) row and 3rd column.

So we multiply the first (only) row of the first matrix by the
3rd column of the second matrix and then add them like this:

{{{4*1 + 1*0 + 0*1 + 2*0 = 4 + 0 + 0 + 0 = 4}}}, so
we put {{{red(4)}}} in the third blank of the product matrix:

{{{(matrix(1,4,
4, 1, 0, 2))*

(matrix(4,3,

1, 0, 1,
2, -1, 0,
3,  5, 1,
1,  3, 0))

=

(matrix(1,3,red(8),red(5),red(4)))

}}}```