Prev Section Next 5. Parametric Form of a System Solution We now know that systems can have either no solution, a unique solution, or an infinite solution.

Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of write a matrix vectorform menu items will be cut off due to the narrow screen width.

There will not be a lot of details in this section, nor will we be working large numbers of examples. The first special matrix is the square matrix. In other words, it has the same number of rows as columns.

In a square matrix the diagonal that starts in the upper left and ends in the lower right is often called the main diagonal. The next two special matrices that we want to look at are the zero matrix and the identity matrix.

Here are the general zero and identity matrices. These are matrices that consist of a single column or a single row. Arithmetic We next need to take a look at arithmetic involving matrices.

If it is true, then we can perform the following multiplication. Here are a couple of the entries computed all the way out.

Determinant The next topic that we need to take a look at is the determinant of a matrix. The determinant is actually a function that takes a square matrix and converts it into a number.

The actual formula for the function is somewhat complex and definitely beyond the scope of this review. The main method for computing determinants of any square matrix is called the method of cofactors.

We can give simple formulas for each of these cases. There is an easier way to get the same result. A quicker way of getting the same result is to do the following.

First write down the matrix and tack a copy of the first two columns onto the end as follows. What we do is multiply the entries on each diagonal up and the if the diagonal runs from left to right we add them up and if the diagonal runs from right to left we subtract them.

Here is the work for this matrix. Matrix Inverse Next, we need to take a look at the inverse of a matrix. Example 4 Find the inverse of the following matrix, if it exists. In other words, we want a 1 on the diagonal that starts at the upper left corner and zeroes in all the other entries in the first three columns.

If you think about it, this process is very similar to the process we used in the last section to solve systems, it just goes a little farther. Here is the work for this problem.

Example 5 Find the inverse of the following matrix, provided it exists. However, there is no way to get a 1 in the second entry of the second column that will keep a 0 in the second entry in the first column.Here is the matrix I pulling the information from just in case you need it.

\begin{bmatrix} 1 & 0 & -1\\ 0 & 1 & 1\\ 0 & 0 & 0 \end{bmatrix} This is what I am thinking. \begin{bmatrix} 1\\ 0\\1 \end{bmatrix} \begin{bmatrix} 0\\ -1\\1 \end{bmatrix} This is my second general . A matrix formulation of the multiple regression model.

as well as learn some of the more important multiple regression formulas in matrix form. As always, let's start with the simple case first. Consider the following simple linear regression function: Since the vector of .

An elementary matrix is a square matrix with one arbitrary column, but otherwise ones along the diagonal and zeros elsewhere (i.e., an identify matrix with the exception of one column).

A.3 Linear Programming in Matrix Form Represent systems of two linear equations with matrix equations by determining A and b in the matrix equation A*x=b.

Problem Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). How to Write a System in Matrix Form. In a system of linear equations, where each equation is in the form Ax + By + Cz + = K, you can represent the coefficients of this system in matrix, called the coefficient matrix.

If all the variables line up with one another vertically, then the first column of the coefficient matrix is dedicated.

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Row and column vectors - Wikipedia