Multiplying A Matrix And A Vector

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Multiplying A Matrix And A Vector

        Multiplying a matrix and a vector is a special case of matrix multiplication.  Circuit equations and state equations representing linear system dynamics contain products of a matrix and a vector.  In the first lesson on circuit analysis, equations that come about by writing node equations can be put into a vector-matrix representation that includes a term that is a matrix - the conductance matrix - multiplied by a vector - the vector of node voltages.  (Click here to go to that point in the lessons where that is presented.)

        Since vector-matrix representations are encountered often in electrical engineering, you need to be very familiar with basic operations.  In this lesson, we will examine mutiplying a matrix and a vector.  In the basic lesson on circuits, we encountered this vector-matrix representation for the circuit below.


        The form we are interested in is this.  We want to be able to evaluate a matrix vector product of this form whenever we encounter one.

        The algorithm for computing the product is best presented visually.  Here it is.

        There are some things to remember about matrix-vector multiplication.

        It is possible to express the calculations mathematically.         Then, the calculation for the the terms in the product vector are given by:

This expression just puts the process for calculating the product into standard mathematical form.  What it says to do is the following.

So, now you should be able to perform these calculations.  Let's look at some example problems.
Questions

Q1. For this matrix and vector, is the product defined?


Q2. For this matrix and vector, is the product defined?


Problems

P1  In this matrix-vector product, the result is a vector, c, with two components, c1 and c2.  Calculate the components of the product and enter your answer in the spaces below.

First, calculate the value of c1
Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:



P1  Next, calculate the value of c2
Enter your answer in the box below, then click the button to submit your answer.

Your grade is:


        Finally,  we have a calculator you can use to avoid doing these kinds of problems by hand.  Here is the calculator.  Here's how to use the calculator.



Problems
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