We cannot simply multiply matrices the way we add and scalar multiply.
The dot product combines two matrices, not necessarily of the same dimension into a third matrix. Understand the location of each element will help you understand the location of the final dot products.
Dimensions Determine the Possibility of a Dot Product
First, we must determine whether the dot product of a matrix is possible.
The dimension of the columns of the first matrix needs to be identical to the dimension of the rows of the second matrix. The simplest way to determine this is either visually (as can be seen above) or writing the dimensions in order. Notice 3X3 and 3X1 both have a 3 in the middle. These matching 3’s indicate we are allowed to perform the dot product operation. The final product will the the outside dimensions, here we have a 3X1 final matrix.
The following is a visual of matching dimensions and the final product of a dot product.
Dot Product as an Operation
Once we determine that the dimensions create a viable situation, we must use our knowledge of the location of the elements to determine how to complete the operation. I find it easiest to trace the progress with my fingers to ensure I don’t skip any elements.
Notice, in the above example, we are determining the first row, first column element, here 58. So we must combine all the elements from the first row from the first matrix to all the elements in the first column from the second matrix. Namely,
We continue with this process to find each of the remaining elements in the “answer” matrix.
First row, second column in the answer will be the first row in the first matrix and the second column in the second matrix.
This process continues to complete the entire matrix.
Here is another example:
It is important to account for every element or you may arrive at an incorrect example. As you get better at the dot product, you will not need to show any work, however anyone new to the process is advised to show the steps until mastery is achieved.
The Inverse of a Matrix
Remember when we defined inverses for real numbers? The product of any number (except zero) and its multiplicative inverse is 1. For example, 2* 1/2=1. In the real number system, the multiplicative inverse in simply the reciprocal. With matrices, the inverse is a little more in depth, however its properties are the same.
Solving Systems of Equations Using Matrices
Generally, matrices are useful for finding the solution to more complex systems of equations. We utilize the inverse of a matrix to quickly compute even the most difficult systems regardless of the number of variables or dimensions. This application of matrices is evident in many other fields beyond mathematics, however, we will stick to a simple system for this course to get the point across.
We can set up our matrices as shown below and use the simple process of multiplying both sides of the equation by the inverse to determine the solution set.
As you can see this process seems excessive and lengthy for a system of two linear equations, however as the systems get more complex and additional variables are added, the simplicity of matrices supersedes any algebraic approach.