Matrix Addition
Purplemath
What is the rule for adding matrices?
To be able to add two matrices, the matrices must be the same size (that is, they must be of the same "order"). This means that they must have the same numbers of rows and columns. So, for instance, a 2 × 3 matrix can be added to another 2 × 3 matrix; it could *not* be added to a 3 × 2 matrix.
How do you add two matrices?
To add matrices, you add corresponding entries of the two matrices. That is, you take the first row, first column entry of the one matrix, and add this to the first row, first column entry of the other matrix. This creates the first row, first column entry in the new matrix. Then you take another entry (say, the first row, second column entry) from the one matrix, add it to the corresponding entry in the other matrix, and make this the value of the first row, second column entry of the sum matrix. And so forth.
This explains why each of the two added matrices must have entries that match up, why two matrices can be added to each other *only* if they have the same numbers of rows and of columns. One matrix can be added to another matrix *only* if they have the same dimensions.
If you're ever given two matrices that have different dimensions, you can immediately conclude that they cannot be added (or subtracted) together.
- Complete the following addition:
![[[ 0 1 2 ][ 9 8 7 ]] + [[ 6 5 4 ][ 3 4 5 ]]](matrices/add01.gif)
In order to add these matrices, I need to add the pairs of entries, and then simplify the sums in order to complete the addition. The first entry of the first row of the first matrix gets added to the first entry of the first row of the second matrix, becoming the value of the first entry of the first row of the sum matrix; and so forth:
![[[ 0 1 2 ][ 9 8 7 ]] + [[ 6 5 4 ][ 3 4 5 ]] = [[ 6 6 6 ][ 12 12 12 ]]](matrices/add02.png)
This final matrix, containing the sums of the entries of the two given matrices, is what they're looking for. So my answer is:
![[[ 6 6 6 ][ 12 12 12 ]]](matrices/add03.gif){kind=small}
And that's really all there is to matrix addition: sum the matching entries, and simplify to get a new matrix (which will be the same size as the two original matrices).
However, while adding (and subtracting) matrices is fairly easy, the fact that you can only add certain matrices to each other — that is, the fact that there are restrictions on adding — is a big deal.
Why is matrix addition a big deal?
Up until now, you've been able to add any two things you felt like: numbers, variables, equations, and so forth. But addition doesn't always work with matrices. As mentioned above, if two matrices are of different dimensions, then there will be matrix entries from the one matrix which have no matching entry in the other matrix. So you can only add matrices that have the same size.
Remember all those number properties you learned about ages ago? Commutative, associative, and distributive? And you wondered why you had to learn them, when everything always obeyed the properties? Matrices are likely the first time you've seen things for which the properties do *not* work. And that's a big deal.
Note that "the same size" for matrices does *not* mean "the same number of entries". You can't take a two-row, three-column matrix and add it to a three-row, two-column matrix. Yes, each of these matrices has six entries, but they're not in the same places, so they cannot be added together. To be added, the two matrices must have the same number of entries in the same configuration.
- Perform the indicated operation, or explain why it is not possible.
![[[ 1 2 ][ 0 3 ]] + [[ 4 5 6 ][ 7 8 9 ]]](matrices/add04.gif)
Matrices are added entry-wise, so I have to add the 1 and the 4, the 2 and 5, the 0 and the 7, and the 3 and the 8.
But what do I add to the 6 and to the 9? There are no corresponding entries in the first matrix that can be added to these entries in the second matrix.
So my answer is:
not possible: matrices aren't the same size
The matrix addition that they were asking me to do is not defined; that is, the attempt at addition doesn't make mathematical sense. So it cannot be done.
Matrix subtraction works entry-wise, too.
- Given the following matrices, find A − B and A − C, or explain why you can not.
![A = [[ -1 2 0 ][ 0 3 6 ]] B = [[ 0 -4 3 ][ 9 -4 -3 ]] C = [[ 0 -4 ][ 9 -4 ]]](matrices/add05.gif)
A and B are the same size, each being 2 × 3 matrices (that is, each matrix has two rows and three columns), so I can subtract them, working entry-wise:
![A - B = [[ -1 2 0 ][ 0 3 6 ]] - [[ 0 -4 3 ][ 9 -4 -3 ]] = [[-1-0 2+4 0-3 ][ 0-9 3+4 6+3]]](matrices/add06.png)
However, A and C are not the same size; A has one more column than does C. So this subtraction is not defined.
![A - B = [[-1 6 -3 ][-9 7 9]]](matrices/add07.gif){kind=small}
A − C not defined: A, B different sizes
Matrix addition and subtraction, where defined (that is, where the matrices are the same size so addition and subtraction make sense), can be turned into homework problems.
- Find the values of xand y, given the following matrix equation:
![[[-3 x ][ 2y 0 ]] + [[ 4 6 ][-3 1 ]] = [[ 1 7 ][-5 1]]](matrices/add08.gif)
First, I'll simplify the left-hand side a bit by adding entry-wise:
![[[-3 x ][ 2y 0 ]] + [[ 4 6 ][-3 1 ]] = [[-3+4 x+6 ][ 2y-3 0+1 ]] = [[ 1 7 ][-5 1]]](matrices/add09.png)
Since matrix equality works entry-wise, I can compare the entries to create simple equations that I can solve. In this case, the 1,2-entries (that is, the entries in the first row, second column of the summed and solution matrices) tell me that x + 6 = 7, and the 2,1-entries (that is, the entries in the second row, first column of the summed and solution matrices) tell me that 2y − 3 = −5.
Solving, I get:
x + 6 = 7
x = 1
2y − 3 = −5
2y = −2
y = −1
These two values are the info they're wanting, so my answer is:
x = 1, y = −1
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