Notes

Matrix Multiplication For Dummies

Daniel Weibel
Created 16 Jul 2016

How To Multiply Two Matrices?

Matrix Multiplication

$m\times n$ matrix:

  • $m$: number of rows (nrow)
  • $n$: number of columns (ncol)

Multiplication of two matrices $m_1 \times n_1$ and $m_2 \times n_2$ is possible if either:

  • $m_1 = n_2$ (i.e. nrow of 1st matrix = ncol of 2nd matrix)
  • $n_1 = m_2$ (i.e. ncol of 1st matrix = nrow of 2nd matrix)

The resulting matrix always has:

  • $m_1$ rows
  • $n_2$ columns

1x4 and 4x2 Matrix

1x4
4x2 $\rightarrow$ OK

                        1  5    4x2 matrix
                        4  3
                        3  1
                        3  2
                      +------
1x4 matrix    4 2 5 2 |33 35    ==> Result: 1x2 matrix

4x1 and 4x2 Matrix

4x1
4x2 $\rightarrow$ Nope

                  1  5    4x2 matrix
                  4  3
                  3  1
                  3  2
                +------
4x1 matrix    4 |
              2 |
              5 |
              2 |         ==> Does not work!

1x4 and 2x4 Matrix

1x4
2x4 $\rightarrow$ Nope

                        1  4  3  3    2x4 matrix
                        5  3  1  2
                      +------------
1x4 matrix    4 2 5 2 |               ==> Does not work!

4x1 and 2x4 Matrix

4x1
2x4 $\rightarrow$ OK

                  1  4  3  3    2x4 matrix
                  5  3  1  2
                +------------
4x1 matrix    4 |24 28 16 20
              2 |12 14  8 10
              5 |30 35 20 25
              2 |12 14  8 10    ==> Result: 4x4 matrix

\hfil \textit{Swap first and second matrix of above multiplications.} \hfil

4x2 and 1x4 Matrix

4x2
1x4 $\rightarrow$ OK

                     4  2  5  2    1x4 matrix
                   +------------
4x2 matrix    1  5 |24 12 30 12
              4  3 |28 14 35 14
              3  1 |16  8 20  8
              3  2 |20 10 25 10    ==> Result 4x4 matrix

4x2 and 4x1 Matrix

4x2
4x1 $\rightarrow$ Nope

                     4    4x1 matrix
                     2
                     5
                     2
                   +---
4x2 matrix    1  5 |
              4  3 |
              3  1 |
              3  2 |      ==> Does not work!

2x4 and 1x4 Matrix

2x4
1x4 $\rightarrow$ Nope

                           4  2  5  2    1x4 matrix
                         +------------
2x4 matrix    1  4  3  3 |
              5  3  1  2 |               ==> Does not work!

2x4 and 4x1 Matrix

2x4
4x1 $\rightarrow$ OK

                           4    4x1 matrix
                           2
                           5
                           2
                         +---
2x4 matrix    1  4  3  3 |33
              5  3  1  2 |35    ==> Result: 2x1 matrix