Understanding Basic Matrix Operations

In this video we are going to discuss matrices, and specifically basic matrix operations focusing on addition and subtraction of matrices. This will cover pages two through four of your lecture notes. First, I said we were going to talk about addition and subtraction of matrices. We want to begin with the definition of addition when it comes to two matrices. If I add two matrices, they must be the same size matrices. So A and B must have the same dimensions. And the resulting matrix when you add, will have the same dimensions as the two original matrices. So that sum of the two matrices will be found by adding corresponding entries. This is why it's important that they have the same size, is so that you do have corresponding entries in order to combine. The result as I said, when you add two matrices that are the same size will be a matrix of the same size as the two original matrices, and we find that resulting matrix by adding corresponding entries of the original two. Subtraction is similar. When we look for or try to find the difference between two matrices, we are going to focus on looking at matrices that are the same size, again because we would like to subtract those corresponding entries. The only thing that is important here is that you make sure that you have your subtraction in the correct order. If I have A minus B, the result may be different than if I have B minus A. But whatever that is, the result will be the same size matrix as the original matrix that you began with. So, when I subtract two matrices I subtract their corresponding entries and that produces the resulting matrix. Let's look at an example. In this example, we are actually going to find three different additions or subtractions. The first is we would like to find the difference between D and C. So we want D minus C. In the second, we would like C minus B. In the third, we would like A plus B. I've given you the following matrices. And when we look at each operation, our first step is going to be to determine if the matrices are the same size. Because remember, addition and subtraction between matrices is only possible if the matrices are the same size. So to begin with part a, we would like to find the difference between matrix D and matrix C. We will begin by determining the size of each matrix. As we can see, matrix D has two rows and three columns. Therefore, matrix D is a two by three matrix. On the other hand, C again also has two rows and three columns, so it is a two by three matrix. Since both matrices are the same size I know I can compute the resulting matrix. The resulting matrix will be a two by three matrix. And in order to compute each entry in the resulting matrix we will take the entry from D minus the entry from C. So the first entry of D was six and we are going to subtract from that three from the first row first column of C. Then our next entry is going to be negative two minus six. And our last entry will be twelve minus negative nine. Then we move to the second row and again, when we subtract the corresponding entries we will begin with D which is negative eight minus the second row first column entry of C which is five. Then three minus seven. And last we will have four minus a negative one. At this point we are ready to do the actual subtraction to compute our resulting matrix. So six minus three gives me three. Negative two minus six is a negative eight. Twelve minus a negative nine, remember order of operations and when you have a negative and a negative that becomes a positive. So we get a positive 21. And moving to the second row I will negative eight minus five which gives me negative 13. Three minus seven which gives me negative four. And four minus a negative one which gives me a positive five. So the resulting matrix of D minus C is three, negative eight, 21, then the second row is negative 13, negative four, positive five. And notice that we end up with a two by three matrix as our final result. Well now that we have taken a look at how to do the subtraction of D minus C by hand, let's talk about using the calculator. Since all entries of my matrices, both D and C contained numbers rather than any variables I can use my calculator to do that simple subtraction. Let's look at how. We begin by going to our calculator. In our calculator we are going to go to the second button and then the x raised to the negative one button which is also our way to get to our matrix list. We have the following menus under our matrices. Names is our first list, that gives us all of the names of the matrices once we want to use them in some sort of computation. Our next menu is all of the math that we can do with the matrices. And our last menu is how we can edit the matrix or enter a matrix into our calculator. So we would like to enter both matrix D and matrix C into our calculator. To help us I will use matrix D and matrix C from our list, but you can put these in any two matrices that you would like. I'm going to select option four for matrix D. Now I need to tell the calculator what size matrix D was. Matrix D was a two by three matrix. And if I press enter after the three, my two by three matrix will appear. Some of your calculators might look slightly different depending on the age of the calculator I will enter the following entries. So matrix D was 6. After each entry I press enter. To get the negative two, I will use the negative symbol to the left of the enter button, not the subtraction symbol. And then twelve was the third entry in that first row. I move to the second row and the calculator does so automatically. And I press the negative eight, then three and four. Once you get to the last entry depending on the age of your calculator, it will entry sit at the last entry or as mine did it will move to the first entry. But it does not go outside that matrix. So you must use second quit so that you don't accidentally put any additional things in the matrix that you did not mean to. We will go back to enter matrix C, again going to second and the matrix option which is the x raised to the negative one button. We will move over to edit and we will go to option three this time. I'll simply press the three. Again I need to tell the calculator what size matrix I have. I have a two by three matrix. If by chance you already have a matrix in your calculator, you can just write over the previous entries and the previous size to change the matrix. I have three, six, negative nine, five, seven, and negative one. I double check to make sure all of my entries are correct in my matrix before I move on. Again I will quit in order to get out to the main screen and not accidentally enter something into my matrix that I did not mean to. Now I'm ready for the actual subtraction. We will go back to our matrix menu. I will start with matrix D. This time I'm not going over to the edit, I stay in the names column. And I'm going to press four. And notice that D appears on my home screen. I will press the subtraction key in order to get the minus and then I go back to my matrices, and I choose option three because I want matrix C Press enter. And you will see that you get the same resulting matrix that we had in our previous example when we did this by hand. For the next part, we want to look at matrix C minus matrix B. Again we're going to start by looking at: are the matrices the same size? Matrix C was a two by three matrix, because it had two rows and three columns. On the other hand, matrix B is a two by two matrix because it has two rows and two columns. Note that while I have the same number of rows I do not have the same number of columns. Therefore, C minus B is not possible as the matrices are not the same size. And we're done, there's nothing else we have to do at this point because my matrices were not the same size, I just state that the operation of subtraction was not possible. In part c, we would like to look at: is the operation of A plus B possible? And if so, what is the resulting matrix? A is we can see is a two by two matrix because it has two rows and two columns. B is also a two by two matrix it has two rows and two columns. Therefore, we know that the operation of A plus B is possible and the resulting matrix should also be a two by two matrix. Let's take a look. So my resulting matrix should be a two by two matrix and I add corresponding entries. So, the first entry of matrix A is negative two and I would like to add positive one from the second matrix, first row, first column entry. Then as we move to the next entry, we have four x as our entry from matrix A. And from matrix B, I want to add negative 11. Moving to the second row, we have that A is y and we want to add three y. And last, our second row, second column entry will be made up of eight plus 18. When I combine, the resulting matrix is: negative two plus one gives me negative one. Four x plus a negative 11 can simply become four x minus 11. Some of you will prefer to put that in parenthesis to show that it is one entry. Then y plus three y becomes four y. And eight plus 18 becomes 26. So the resulting matrix for A plus B is negative one, four x minus 11. The second row is four y, 26. And again note that the result was a two by two matrix. One thing that is important to note here is that my matrices A and B contain variables. Therefore, we are unable to use our calculator because there are variable entries. Anytime you have variable entries, you will have to do the operations by hand. Next we would like to look at some properties of matrix addition specifically. These properties may or may not apply to subtraction, so it's important that you think of them only in terms of addition. So the first property is the commutative property. And the commutative property says: as long as all of your matrices are the same size, the order in which you add the matrices does not matter. So I can have A plus B, and the resulting matrix would be the same as if I added B plus A. Again, this is only true for addition. It is not true for subtraction. Next we have the associative property. And the associative property, again if all the matrices are the same size, I can either first add matrix A and matrix B, and then take the result and add matrix C. Or I can add matrix B and matrix C, and then take the result and add that to matrix A. And again the two sides would be the same. The resulting matrices would be equal. Next we have the identity property. The identity property for addition says: if you have a matrix that only contains zeros as all of its entries, and it is a matrix of the same size as matrix A, then when you add the matrix and the identity matrix, or the zero matrix of the same size, the result will always be the original matrix A. And last we have the inverse property. And the inverse property for addition when it comes to matrices is: if I have a matrix, then its inverse or its additive inverse to be more specific is all entries with opposite signs and the original matrix. So you take the original matrix, change all of the signs, and the resulting matrix is the additive inverse. When you add the two matrices together, the resulting matrix will be the zero matrix of the same size as the original matrix A. Our additive inverse is the same size as matrix A as well. Let's take a look at two examples. So here I have the same four matrices we had in example two. And first we want to find the additive inverse of matrix B, if possible. And then second, we want to state the size of the additive identity matrix for matrix D. Let's begin with part a. In part a, we would like to state the additive inverse of matrix B. If we look at matrix B, remember the additive inverse is the matrix with all opposite signs. So I look at matrix B and change the sign of every entry. I began with a positive one, so that becomes a negative one. Next we had a negative 11 which becomes a positive 11. I don't have to have the plus sign but that's the emphasize that I am changing the sign. But it is not required. Then you have three y and the opposite of three y will be negative three y. And last, we have a positive 18 and that becomes a negative 18. The additive inverse of matrix B, which we denote with the negative symbol in front of the matrix name, so the additive inverse of matrix B will be negative one, positive 11, negative three y, negative 18. In part b, we want to find the size of the additive identity of matrix D. Recall that the additive identity for a matrix is a matrix of the same size as the matrix that we are looking at. D, we said earlier, is a two by three matrix. Therefore, our identity, so our additive identity matrix will also be a two by three matrix. It is the same size as the original matrix given. So the original matrix was a two by three making our additive identity matrix a two by three matrix.

In this video we are going to discuss matrices, and specifically basic matrix operations focusing on addition and subtraction of matrices.  This will cover pages two through four of your lecture notes. First, I said we were going to talk about addition and subtraction of matrices. We want to begin with the definition of addition when it comes to two matrices.  If I add two matrices, they must be the same size matrices. So A and B must have the same dimensions. And the resulting matrix when you add, will have the same dimensions as the two original matrices.  So that sum of the two matrices will be found by adding corresponding entries. This is why it's important that they have the same size, is so that you do have corresponding entries in order to combine.  The result as I said, when you add two matrices that are the same size will be a matrix of the same size as the two original matrices, and we find that resulting matrix by adding corresponding entries of the original two.  Subtraction is similar. When we look for or try to find the difference between two matrices, we are going to focus on looking at matrices that are the same size, again because we would like to subtract those corresponding entries.  The only thing that is important here is that you make sure that you have your subtraction in the correct order. If I have A minus B, the result may be different than if I have B minus A. But whatever that is, the result will be the same size matrix as the original matrix that you began with.  So, when I subtract two matrices I subtract their corresponding entries and that produces the resulting matrix. Let's look at an example. In this example, we are actually going to find three different additions or subtractions.  The first is we would like to find the difference between D and C. So we want D minus C. In the second, we would like C minus B. In the third, we would like A plus B.  I've given you the following matrices. And when we look at each operation, our first step is going to be to determine if the matrices are the same size. Because remember, addition and subtraction between matrices is only possible if the matrices are the same size.  So to begin with part a, we would like to find the difference between matrix D and matrix C. We will begin by determining the size of each matrix. As we can see, matrix D has two rows and three columns. Therefore, matrix D is a two by three matrix.  On the other hand, C again also has two rows and three columns, so it is a two by three matrix. Since both matrices are the same size I know I can compute the resulting matrix.  The resulting matrix will be a two by three matrix. And in order to compute each entry in the resulting matrix we will take the entry from D minus the entry from C. So the first entry of D was six and we are going to subtract from that three from the first row first column of C.  Then our next entry is going to be negative two minus six. And our last entry will be twelve minus negative nine. Then we move to the second row and again, when we subtract the corresponding entries we will begin with D which is negative eight minus the second row first column entry of C which is five. Then three minus seven. And last we will have four minus a negative one.  At this point we are ready to do the actual subtraction to compute our resulting matrix. So six minus three gives me three. Negative two minus six is a negative eight. Twelve minus a negative nine, remember order of operations and when you have a negative and a negative that becomes a positive. So we get a positive 21.  And moving to the second row I will negative eight minus five which gives me negative 13. Three minus seven which gives me negative four.  And four minus a negative one which gives me a positive five. So the resulting matrix of D minus C is three, negative eight, 21, then the second row is negative 13, negative four, positive five. And notice that we end up with a two by three matrix as our final result.   Well now that we have taken a look at how to do the subtraction of D minus C by hand, let's talk about using the calculator. Since all entries of my matrices, both D and C contained numbers rather than any variables I can use my  calculator to do that simple subtraction. Let's look at how. We begin by going to our calculator. In our calculator we are going to go to  the second button and then the x raised to the negative one button which is also our way to get to our matrix list.  We have the following menus under our matrices. Names is our first list, that gives us all of the names of the matrices once we want to use them in some sort of computation.  Our next menu is all of the math that we can do with the matrices. And our last menu is how we can edit the matrix or enter a matrix into our calculator.  So we would like to enter both matrix D and matrix C into our calculator. To help us I will use matrix D and matrix C from our list, but you can put these in any two matrices that you would like. I'm going to select option four for matrix D. Now I need to tell the calculator what size matrix D was. Matrix D was a two by three matrix. And if I press enter after the three, my two by three matrix will appear. Some of your calculators might look slightly different depending on the age of the calculator I will enter the following entries. So matrix D was 6. After each entry I press enter. To get the negative two, I will use the negative symbol to the left of the enter button, not the subtraction symbol.  And then twelve was the third entry in that first row. I move to the second row and the calculator does so automatically. And I press the negative eight, then three and four.  Once you get to the last entry depending on the age of your calculator, it will entry sit at the last entry or as mine did it will move to the first entry. But it does not go outside that matrix.  So you must use second quit so that you don't accidentally put any additional things in the matrix that you did not mean to. We will go back to enter matrix C, again going to second and the matrix option which is the x raised to the negative one button.  We will move over to edit and we will go to option three this time. I'll simply press the three. Again I need to tell the calculator what size matrix I have. I have a two by three matrix. If by chance you already have a matrix in your calculator,  you can just write over the previous entries and the previous size to change the matrix.  I have three, six, negative nine, five, seven, and negative one. I double check to make sure all of my entries are correct in my matrix before I move on.  Again I will quit in order to get out to the main screen and not accidentally enter something into my matrix that I did not mean to.  Now I'm ready for the actual subtraction. We will go back to our matrix menu. I will start with matrix D. This time I'm not going over to the edit, I stay in the names column.  And I'm going to press four. And notice that D appears on my home screen. I will press the subtraction key in order to get the minus and then I go back to my matrices, and I choose option three because I want matrix C Press enter. And you will see that you get the same resulting matrix that we had in our previous example when we did this by hand.  For the next part, we want to look at matrix C minus matrix B. Again we're going to start by looking at: are the matrices the same size?  Matrix C was a two by three matrix, because it had two rows and three columns. On the other hand, matrix B is a two by two matrix because it has two rows and two columns.  Note that while I have the same number of rows I do not have the same number of columns. Therefore, C minus B is not possible as the matrices are not the same size.    And we're done, there's nothing else we have to do at this point because my matrices were not the same size, I just state that the operation of subtraction was not possible.  In part c, we would like to look at: is the operation of A plus B possible? And if so, what is the resulting matrix?  A is we can see is a two by two matrix because it has two rows and two columns. B is also a two by two matrix it has two rows and two columns.  Therefore, we know that the operation of A plus B is possible and the resulting matrix should also be a two by two matrix. Let's take a look.  So my resulting matrix should be a two by two matrix and I add corresponding entries. So, the first entry of matrix A is negative two and I would like to add positive one from the second matrix, first row, first column entry. Then as we move to the next entry, we have four x as our entry from matrix A. And from matrix B, I want to add negative 11. Moving to the second row, we have that A is y and we want to add three y. And last, our second row, second column entry will be made up of eight plus 18.  When I combine, the resulting matrix is: negative two plus one gives me negative one.  Four x plus a negative 11 can simply become four x minus 11.  Some of you will prefer to put that in parenthesis to show that it is one entry.  Then y plus three y becomes four y. And eight plus 18 becomes 26. So the resulting matrix for A plus B is negative one, four x minus 11. The second row is four y, 26.  And again note that the result was a two by two matrix.  One thing that is important to note here is that my matrices A and B contain variables. Therefore, we are unable to use our calculator because there are variable entries.  Anytime you have variable entries, you will have to do the operations by hand.  Next we would like to look at some properties of matrix addition specifically.  These properties may or may not apply to subtraction, so it's important that you think of them only in terms of addition. So the first property is the commutative property.  And the commutative property says: as long as all of your matrices are the same size, the order in which you add the matrices does not matter.  So I can have A plus B, and the resulting matrix would be the same as if I added B plus A.  Again, this is only true for addition. It is not true for subtraction.  Next we have the associative property. And the associative property, again if all the matrices are the same size, I can either first add matrix A and matrix B, and then take the result and add matrix C. Or I can add matrix B and matrix C, and then take the result and add that to matrix A. And again the two sides would be the same. The resulting matrices would be equal.  Next we have the identity property. The identity property for addition says: if you have a matrix that only contains zeros as all of its entries, and it is a matrix of the same size as matrix A,  then when you add the matrix and the identity matrix, or the zero matrix of the same size, the result will always be the original matrix A.  And last we have the inverse property. And the inverse property for addition when it comes to matrices is: if I have a matrix, then its inverse or its additive inverse to be more specific  is all entries with opposite signs and the original matrix. So you take the original matrix, change all of the signs, and the resulting matrix is the additive inverse.  When you add the two matrices together, the resulting matrix will be the zero matrix of the same size as the original matrix A. Our additive inverse is the same size as matrix A as well. Let's take a look at two examples.  So here I have the same four matrices we had in example two. And first we want to find the additive inverse of matrix B, if possible.  And then second, we want to state the size of the additive identity matrix for matrix D. Let's begin with part a.  In part a, we would like to state the additive inverse of matrix B.  If we look at matrix B, remember the additive inverse is the matrix with all opposite signs. So I look at matrix B and change the sign of every entry.  I began with a positive one, so that becomes a negative one. Next we had a negative 11 which becomes a positive 11. I don't have to have the plus sign but that's the emphasize that I am changing the sign. But it is not required. Then you have three y and the opposite of three y will be negative three y. And last, we have a positive 18 and that becomes a negative 18. The additive inverse of matrix B,  which we denote with the negative symbol in front of the matrix name, so the additive inverse of matrix B will be negative one, positive 11, negative three y, negative 18.  In part b, we want to find the size of the additive identity of matrix D. Recall that the additive identity for a matrix is a matrix of the same size as the matrix that we are looking at.  D, we said earlier, is a two by three matrix. Therefore, our identity, so our additive identity matrix will also be a two by three matrix.  It is the same size as the original matrix given. So the original matrix was a two by three making our additive identity matrix a two by three matrix.

Transcript for:Understanding Basic Matrix Operations

Transcript for:
Understanding Basic Matrix Operations