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## HP 40gs Graphing Calculator - Using Matrices

How are matrices stored?
Matrices are stored and edited in the Matrix Catalog. It can hold matrices that are vectors or contain complex numbers. Once defined, a matrix or vector can be used or manipulated in any other view. Some extremely powerful matrix function are included in the MATH menu. It can be cleared using SHIFT CLEAR.
Figure : Matrix Catalog
Given that and , find the value of B-1A.
Enter the Matrix Catalog by pressing SHIFT MATRIX. With the highlight on M1, press EDIT. Enter the values 2 and 3, pressing ENTER after each. Press down arrow to begin a new line and enter the values –1 and 4. Leave the edit view by pressing SHIFT MATRIX again. Use the same method to create M2.
Figure : Creating the first matrix
Figure : Entering the values of the first matrix
Now in the HOME view, perform the calculation by typing ALPHA M 2 X-1* ALPHA M1. Then store the solution in M3 in case it is required later. The value displayed can be viewed more easily using SHOW.
Figure : HOME view
Figure : Displaying the final answer
How do I solve a system of equations?
Systems of simultaneous linear equations of any size can be solved either with an inverse matrix or using the function RREF from the MATH menu.
The sales of type A, B and C computers for three successive weeks are shown below. Find the prices of each type of computer.
This system of equations can be represented in matrix form as:
If we enter the first matrix as M1 and the values as M3 then the solution is M1-1*M3.
Change to the Matrix Catalog and enter the M1 as outlined on the previous page. Before entering the values into M2, press GO→ to change it to GO↓ . This moves the cursor down instead of right after each entry.
Figure : Values of M1
Figure : Values of M2
Rounding error may make it difficult to see the solution in the HOME view. Viewing M3 is probably the better option.
Figure : HOME view
Figure : Viewing elements of M3
If there is no valid solution then the matrix inverse will not exist, as shown below.
Figure : Invalid solution
Here it is better to use the RREF function (Reduced Row Echelon Form). It acts on an augmented matrix and its advantage is that it will work even if the equations are inconsistent. Three cases can result, as below.
Case 1: A unique solution
becomes
Figure : Display of augmented matrix M1
The final column of the matrix contains the solution.
Figure : Using the RREF function
Figure : Display on using the RREF function
Case 2: No solution
becomes
Figure : Display of augmented matrix M1
In this case the final line of the reduced row echelon matrix is [ 0 0 0 1 ], hence no solution.
Figure : Using the RREF function
Figure : Display on using the RREF function
Case 3: Infinite solutions
This case is similar to that of no solution but the final line of the reduced row echelon matrix will be [ 0 0 0 0 ], which corresponds to the case of infinite solutions
Quick and easy roots of a polynomial
Although it is possible to find roots of any graph by using the PLOT view and FCN, the MATH menu provides a function called POLYROOT which will find all the roots of any polynomial in one operation.
Find the roots of f(x) = x4 - 27x2 - 14x + 135.
Figure : Using the Polyroot function
Coefficients must be supplied in the form of a row vector using square brackets and solutions are returned the same form. If any of the roots are complex then the entire set will be returned in complex form (a,b) as in the example below.
Figure : Displaying the roots of f(x)
Figure : Displaying the roots of f(x)
##### note:
A worthwhile tip here is to store the solution vector into a matrix variable. This allows easy viewing of the solutions, both real and complex.

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