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What is the Identity Property of Matrix Addition?

What's the Identity Property of Matrix Addition?

Summary

  1. 'a' stands for any number
  2. 0 is the additive identity because adding it to a number gives you the same number back
  3. A zero matrix is a matrix whose elements are all zeros
  4. 'a', 'b', 'c', and 'd' stand for numbers in each element of the matrix
  5. 'A' represents any matrix
  6. The '0' in the general property stands for the zero matrix
  7. Because of the Commutative Property of Matrix Addition, you can add the two matrices in any order and still get the same thing

Notes

    1. The Identity Property of Addition says that if you add 0 to a number, then that number stays the same
    2. 0 is called the 'additive identity'
    1. In other words, adding 0 to a number does not change anything
    2. 'a' represents any number
    3. 0 is the additive identity
    1. Let's see if there is an identity property for matrices as well
    1. Our example matrix is a 2x2, or two by two matrix
    1. A zero matrix is a matrix whose elements are all zeros
    2. Zero matrices can have any dimensions, as long as all the elements are zero
    3. Here we need a zero matrix with the same dimensions as our original matrix, otherwise we can't add them
    4. So we'll make our zero matrix 2x2 as well
    1. The Identity Property of Addition says that you can add zero to any number and you'll always get that number back
    2. We can see that happens here with each element in our new matrix!
    1. Now that we've seen that the identity property works for one matrix, let's see if it works in general
    1. Each variable, 'a', 'b', 'c', and 'd', stands for any number
    2. Since we have variables in this new matrix, whatever we find out about adding the zero matrix will apply to any other matrix
    1. A zero matrix is a matrix whose elements are all zeroes
    2. The original generalized matrix had 'a', 'b', 'c', and 'd' as elements
    3. And remember, if we add 0 to a number we get that same number back
    4. So when we add 0 to each of our variable elements, we get those same variables back as the elements of our new matrix!
    1. Since this worked in general, that means there is an identity property for matrices as well!
    1. A zero matrix is a matrix whose elements are all zeroes
    2. Make sure the zero matrix has the same dimensions as 'A', otherwise you can't add them!
    3. Adding the zero matrix will not change the original matrix, since all we're doing is adding 0 to each element!