6.05 Matrix multiplication

Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second.

If matrix A has dimension m \times n and matrix B as dimension a\times b then the multiplication is defined (possible) if n=a and the dimension of the result C=AB will be m \times b. This is shown in the following diagram:

If the dimensions do not meet this criteria then we say that the matrix multiplication is undefined (not possible).

For example consider the following matrices A, B and C where A=\begin{bmatrix} a&b\\ c&d \end{bmatrix}, B=\begin{bmatrix} e\\ f \end{bmatrix}, C=\begin{bmatrix} g&h \end{bmatrix}

A is a 2\times 2 matrix, B is a 2\times 1 matrix and C is a 1\times 2 matrix.

• A\times B is a (2\times 2)\times (2\times 1) this is defined and the result would be a 2\times 1 matrix.

• B\times C is a (2\times 1)\times (1\times 2) this is defined and the result would be a 2\times 2 matrix.

• A\times C is a (2\times 2)\times (1\times 2) this is undefined.

Unlike matrix addition and subtraction, matrix multiplication is not computed element by element.

Let's look at an example of multiplication before we generalise the result.

If we take the elements in the first row, and multiply them by the elements in the first column, then sum them up then we end up with the a number that belongs in the spot occupying the first row and first column.

So 5\times 6+12\times 1+56\times 7=434 goes into the (1,1) entry: \begin{bmatrix} 5&12&56\\ 10&30&75\\ \end{bmatrix} \begin{bmatrix} 6&3\\ 1&2\\ 7&2 \end{bmatrix} = \begin{bmatrix} 434&.\\ .&. \end{bmatrix}

The next combination will be the first row and the second column. Take the sum of the elements in the first row multiplied by the elements in second column.

So, 5\times 3+12\times 2+56\times 2=151 goes into the (1,2) entry: \begin{bmatrix} 5&12&56\\ 10&30&75\\ \end{bmatrix} \begin{bmatrix} 6&3\\ 1&2\\ 7&2 \end{bmatrix} = \begin{bmatrix} 434&151\\ .&. \end{bmatrix}

The next combination will be the second row and the first column. Take the sum of the elements in the second row multiplied by the elements in first column.

So 10\times 6+30\times 1+75\times 7=615 goes into the (2,1) entry: \begin{bmatrix} 5&12&56\\ 10&30&75\\ \end{bmatrix} \begin{bmatrix} 6&3\\ 1&2\\ 7&2 \end{bmatrix} = \begin{bmatrix} 434&151\\ 615&. \end{bmatrix}

The final combination is the second row and the second column. Take the sum of the elements in the second row multiplied by the elements in second column.

So 10\times 3+30\times 2+75\times 2=240 goes into the (2,2) entry. \begin{bmatrix} 5&12&56\\ 10&30&75\\ \end{bmatrix} \begin{bmatrix} 6&3\\ 1&2\\ 7&2 \end{bmatrix} = \begin{bmatrix} 434&151\\ 615&240 \end{bmatrix}

In general, to compute element (i,j) in the matrix AB, we multiply the elements of the ith row in A with the elements in the jth column of B, and sum all the products. For a pair of 2\times 2 matrices, this looks like this: \begin{bmatrix} a&b\\ c&d \end{bmatrix} \begin{bmatrix} e&f\\ g&h \end{bmatrix} =\begin{bmatrix} ae+bg&af+bh\\ ce+dg&cf+dh \end{bmatrix}

Examples

The Identity matrix, I, has similar properties to the number 1 in the real number system. When multiplying a matrix by the identity matrix, I, of the appropriate order, the result is the original matrix. \begin{bmatrix} a&b\\ c&d \end{bmatrix} \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} = \begin{bmatrix} a&b\\ c&d \end{bmatrix}

The diagonal matrix, with 1's on the diagonal is the identity matrix. Use the symbol I, to represent the identity matrix. \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} = I

A special thing about the identity matrix is that the order of multiplication doesn't matter, i.e.

Let A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} and let I= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} then AI=IA=A.

Examples