Matrix: Zero

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We can use the GNRNDM program with the key values -3452801405, 2341433233 and 4413232332 to generate the matrices:

The calculator used to obtain the images below was a TI 84 Plus running the 2.55MP operating system and it was in the MATHPRINT display mode. These images merely reflect generating the matrices. Please note that even though the calculator is in MATHPRINT mode, the programmatic output in these Figures shows up in CLASSIC mode. The nicer display of the MATHPRINT mode shows up after the program completes.

We now have matrices that we can use on this page.

In an earlier page we looked at two properties of addition for matrices, namely, the commutative and associative properties. Another property that we learned when looking at numbers was the addition property of zero. By that we meant that we have a number zero, 0, that we can add to any number and the result is the same value. Thus, 4+0=4 and 0+6=6. Now we are looking at matrices. Is there a zero property for the addition of matrices?

The answer is Yes, but... because there is not just one zero matrix. Instead, we need a zero matrix for every distinct dimension. Thus, there is a 2 x 3 zero matrix and a different 3 x 2 zero matrix. We will examine these below.

Figure 01	is a `3 x 2` matrix. In order to add another matrix to it we need that other matrix to be a `3 x 2` matrix. Therefore, we need to create a `3 x 2` matrix matrix where every element of that matrix has the value `0`. Unfortunately, the TI-83/84 calculators have exactly ten (10) matrices. The GNRNDM program that we ran filled all of those. We will convert [A] to be the desired `3 x 2` matrix zero matrix. Our first step is to change the dimensions of [A]. We start to form the command to do that via the keys as shown in Figure 01. This is a "list" that holds the desired dimensions for our matrix.
Figure 02	We continue the command by finding the appropriate `dim(` code. First we use to open the Matrix window.
Figure 03	The we use to move the highlight to the MATH option, shown in Figure 03 after we have used to put the highlight on the desired choice. At that point we can press to move to Figure 04.
Figure 04	The result shown in Figure 04 needs to be completed. We now need that name of the matrix that we want to use. We return to the Matrix window via (this was shown in Figure 02 where we note that the desired matrix, [A], is the highlighted one) and then use to select the [A] name.
Figure 05	Then we complete the command with and press to have the calculator process the command. The only result that the calculator displays is to echo the list `{3,2}` as shown in Figure 05. The command that we just executed merely changed the dimensions of [A]. Since we had values in the matrix before, the new version of [A] will retain those values as if we had just cut off the last column. (Note that had we expanded the matrix the new positions would have held the value `0`.)
Figure 06	In order to set all of the values in [A] with the value `0` we want to use the Fill( command. We find that in the Matrix menu under the MATH option. As shown in Figure 06, it is the fourth option. Once we have moved the highlight to that option,press to paste it to the main screen.
Figure 07	Now that we have Fill( on the main screen we want to complete the command. We need to give the command the value to use, `0` and the name of the matrix to use, [A].
Figure 08	Figure 08 shows the completed command and the calculator display of performing the command, namely the word Done.
Figure 09	Now, if we return to the matrix menu and select the name [A] to paste onto the main screen and then we press , .the calculator displays the contents of [A]
Figure 10	With [A] in place we can construct the command to add [A] and [F], namely [A]+[F]. Figure 10 shows the result and it is exactly as we expect: our zero `3 x 2` matrix plus another `3 x 2` matrix results in a matrix identical to the original second addend. Just as with numbers `0+4=4` so with our zero matrix we have [A]+[F]=[F].
Figure 11	We can use our zero matrix, [A], with another `3 x 2` matrix as in [A]+[B] as shown in Figure 11.
Figure 12	However, if we try to use [A] with a matrix of different dimensions, say [E] which is a `2 x 3` matrix we expect to run into trouble. Figure 12 shows that command.
Figure 13	Figure 13 shows the resulting error message.
Figure 14	Figure 14 shows the steps to change [A] to a `2 x 3` matrix and then to fill that matrix with 0's.
Figure 15	Once that is done then we can easily find [A]+[E] as is shown in Figure 15.
Figure 16	Of course, since we remember that matrix addition is commutative, we understand that we can add the appropriate zero matrix on either the left or right side. This is demonstrated in Figure 16.