LAB 2: Graphing Calculator: Storing and Recalling Data in a Matrix

New Concept

A graphing calculator can be used to work with matrices.

Why it matters

Matrices provide a compact language for manipulating large sets of data and solving complex systems of equations all at once. Mastering them on your calculator gives you a computational superpower, essential for tackling advanced problems in science and cryptography.

What’s next

Next, you'll learn the keypresses to store, recall, and perform fundamental matrix operations like multiplication and finding the inverse on your calculator.

To store a matrix, navigate to the MATRIX menu, select EDIT, and choose a matrix name like [A]. You must first define the dimensions by entering the number of rows and columns. Then, you can input each element of the matrix one by one, pressing ENTER after each entry to save it.

Example 1: To store $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, go to MATRIX > EDIT > 1:[A], set dimensions to 2x2, and enter the four values.
Example 2: To store $B = \begin{bmatrix} 5 & 0 \\ -2 & 1 \end{bmatrix}$, go to MATRIX > EDIT > 2:[B], set dimensions to 2x2, and input its elements.

Think of your calculator's matrix memory as a set of labeled file folders. Storing a matrix is like creating a new file, giving it a name like 'A', and putting your data inside. Once it's saved, it's ready to be used for any calculation you need, so you don’t have to re-type it every single time.

To perform calculations like $A \times B$, start from the home screen. Access the MATRIX menu and under the NAMES tab, select [A] to place it on the screen. Press the multiplication key, then access the menu again to select [B]. Pressing ENTER will execute the operation and display the resulting matrix.

Example 1: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 0 \\ -2 & 1 \end{bmatrix}$, typing [A] * [B] on the calculator yields $\begin{bmatrix} 1 & 2 \\ 7 & 4 \end{bmatrix}$.
Example 2: For $A = \begin{bmatrix} -2 & 0 \\ 7 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -5 \\ -3 & 0 \end{bmatrix}$, typing [A] * [B] yields $\begin{bmatrix} -2 & 10 \\ -2 & -35 \end{bmatrix}$.

Once your matrices are safely stored, this is the fun part! You just tell the calculator which stored matrices to grab. It's like a chef grabbing prepped ingredients. You call matrix 'A' and matrix 'B' from the menu, tell the calculator to multiply them, and it serves up the final answer instantly without any messy manual calculations.

To find the determinant of a matrix, go to the MATRIX menu and arrow over to the MATH submenu. Select the det( function. Then, go back to the MATRIX > NAMES menu to select the matrix you want, like [A]. Close the parenthesis and press ENTER to calculate the single value determinant.

Example 1: For matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, using det([A]) on the calculator results in $-2$.
Example 2: For matrix $A = \begin{bmatrix} -2 & 0 \\ 7 & 3 \end{bmatrix}$, using det([A]) on the calculator results in $-6$.

The determinant is a special number that tells you important things about a matrix. Instead of a tedious manual calculation, your calculator can find it in seconds! You just wrap the det() function around your stored matrix name. The calculator instantly processes it and gives you that one magical number that represents the matrix's unique properties.

To solve a system of equations, first enter it as an augmented matrix. From the home screen, go to MATRIX > MATH and select the rref( function. Then, from the MATRIX > NAMES menu, choose your augmented matrix. Closing the parenthesis and pressing ENTER will display the fully solved matrix.

Example 1: For $C = \begin{bmatrix} 3 & -2 & 5 \\ -1 & 4 & 7 \end{bmatrix}$, using rref([C]) yields $\begin{bmatrix} 1 & 0 & 3.4 \\ 0 & 1 & 2.6 \end{bmatrix}$.
Example 2: For $C = \begin{bmatrix} -2 & -10 & 1 \\ 1 & 7 & -6 \end{bmatrix}$, using rref([C]) yields $\begin{bmatrix} 1 & 0 & -12.75 \\ 0 & 1 & 2.45 \end{bmatrix}$.

This sounds super complex, but rref( is your best friend for solving systems of equations. You give the calculator a jumbled augmented matrix, and this function works like a super-powered organizer, cleaning it all up into a simple form. The final matrix neatly displays the solutions for each variable, making it one of the most powerful calculator shortcuts!