2.1 Matrix multiplication

A matrix, normal cap a , is a rectangular array of numbers arranged in rows and columns. (Matrices is the plural of the word matrix.) You will concentrate on two-dimensional situations, and so consider matrices of the form:

matrix row 1column 1 cap a 11 cap a 12 row 2column 1 cap a 21 cap a 22 times two multiplication two square matrix

matrix row 1column 1 cap a 11 cap a 12 times one multiplication two row matrix

vector element 1 cap a sub 11 element 2 cap a sub 21 times two multiplication one column matrix

In order to multiply two matrices, the number of columns of first matrix should be equal to the number of rows in the second matrix. Matrices of the right shape can be multiplied together as follows:

normal cap c equals normal cap a times normal cap b full stop

Equation label:(1)

To calculate normal cap c , where normal cap c sub i times j is the element in the i th row and j th column of , you go along the th row of normal cap a and down the th column of normal cap b , multiplying corresponding elements and adding the results. For two two multiplication two matrices, this pattern may be visualized as follows:

matrix row 1column 1 asterisk operator equals matrix row 1column 1 right arrow right arrow times matrix row 1column 1 down arrow row 2column 1 down arrow

where the asterisk operator indicates a matrix element in the new matrix, normal cap c , and the arrows show how matrix elements in the old matrices are processed to obtain this. In order to multiply two matrices, the number of columns of first matrix should be equal to the number of rows in the second matrix. In other words, each term cap c sub i times j is given by cap c sub i times j equals cap a sub i times one times cap b sub one times j plus cap a sub i times two times cap b sub two times j .

Exercise 3

What shape is the result of multiplying a row matrix by a square matrix?
What shape is the result of multiplying a square matrix by a row matrix?
What shape is the result of multiplying a square matrix by a column matrix?
What shape is the result of multiplying a row matrix by a column matrix?

Answer

The result is a row matrix.
This operation cannot be performed because the number of columns of first matrix is not equal to the number of rows in the second matrix.
The result is a column matrix.
The result is a matrix (i.e. a scalar number).

Exercise 4

Evaluate the following combination of square matrices:

matrix row 1column 1 12 row 2column 1 two minus minus one times matrix row 1column 1 10 row 2column 1 zero minus minus one times matrix row 1column 1 12 row 2column 1 two minus minus one

Answer

The product of two matrices is given by:

matrix row 1column 1 cap a 11 cap a 12 row 2column 1 cap a 21 cap a 22 times matrix row 1column 1 cap b 11 cap b 12 row 2column 1 cap b 21 cap b 22 equals matrix row 1column 1 plus plus times times cap a 11 cap b 11 times times cap a 12 cap b 21 plus plus times times cap a 11 cap b 12 times times cap a 12 cap b 22 row 2column 1 plus plus times times cap a 21 cap b 11 times times cap a 22 cap b 21 plus plus times times cap a 21 cap b 12 times times cap a 22 cap b 22

So, to find the matrix element in row and column of the product normal cap a times normal cap b , we multiply the elements in row of with the corresponding elements in column of , and add the results together.

To find the product of three matrices, normal cap a times normal cap b times normal cap c , we first evaluate and then form the product normal cap a of normal cap b times normal cap c , taking care to preserve the order of the matrices.

So the solution is

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