Adding and Subtracting Matrices Matrix addition is fairly simple, and is done entrywise.
I need to add the pairs of entries, and then simplify for the final answer: So the answer is: 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.
Since matrices are added entrywise, 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 the answer is: I can't add these matrices, because they're not the same size. This is always the case: To be able to add two matrices, they must be of the same size. If they are not the same size (if they do not have the same "dimensions"), then the addition is "not defined" (doesn't make mathematical sense). Subtraction works entrywise, too. Copyright © Elizabeth Stapel 20032011 All Rights Reserved
A and B are the same size, each being 2 × 3 matrices, so I can subtract, working entrywise: However, A and C are not the same size, since A is 2 × 3 and C is 2 × 2. So this subtraction is not defined. A – C is not defined, because A and C are not the same size. 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.
First, I'll simplify the lefthand side a bit by adding entrywise: Since matrix equality works entrywise, I can compare the entries to create simple equations that I can solve. In this case, the 1,2entries tell me that x + 6 = 7, and the 2,1entries tell me that 2y – 3 = –5. Solving, I get: x + 6 = 7
2y – 3 = –5



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