## Multiplying Matrices

A matrix multiplied by a another matrix will result in another matrix (through the matrix may actually be a 1 x 1 vector, which is a number). In general an n x m matrix times an m x o matrix will result in an n x o matrix. Each entry of the resulting vector is found by multiplying the entries of each row of the mattrix first by the entries of each column of the second matrix and adding them.

Example: The 2 x 3 matrix is to be multiplied by the 3 x 4 matrix The result will be a 2 x 4 matrix.

The first entry in the resulting matrix will be found by multiplying the entries in the first row of the first matrix by the entries in the first column of the second matrix and adding them. We obtain 2*0+5*1+4*1 = 9.

The second entry (moving along the top row) will be found by multiplying the entries in the first row of the first matrix by the entries in the second column of the second matrix and adding them. We obtain 2*0+5*-2+4*2 = 0.

The third entry will be found by multiplying the entries in the first row of the first matrix by the entries in the third column of the second matrix and adding them. We obtain 2*3+5*1+4*-1 = 7.

The fourth entry will be found by multiplying the entries in the first row of the first matrix by the entries in the fourth column of the second matrix and adding them. We obtain 2*5+5*-4+4*3 = 2.

The first row of the resulting matrix is The first entry in the second row of the resulting matrix will be found by multiplying the entries in the second row of the first matrix by the entries in the first column of the second matrix and adding them. We obtain 1*0+0*1+-3*1 = -3.

The second entry in the second row of the resulting matrix will be found by multiplying the entries in the second row of the first matrix by the entries in the second column of the second matrix and adding them. We obtain 1*1+0*-2+-3*2 = -5.

The third entry in the second row of the resulting matrix will be found by multiplying the entries in the second row of the first matrix by the entries in the third column of the second matrix and adding them. We obtain 1*3+0*1+-3*-1 = 6.

The fourth entry in the second row of the resulting matrix will be found by multiplying the entries in the second row of the first matrix by the entries in the fourth column of the second matrix and adding them. We obtain 1*5+0*-4+-3*3 = -4.

The second row or the resulting matrix is The product of the two matrices is  