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What is the Inverse Property of Matrix Addition?

What is the Inverse Property of Matrix Addition?

Summary

  1. 'a' stands for any number
  2. If you add any number with its opposite, you'll always get zero
  3. A number's opposite is that number multiplied by -1
  4. A matrix's opposite is that matrix multiplied by the scalar -1
  5. 'A' stands for any matrix, and '-A' is its opposite
  6. The bold '0' represents the zero matrix, which will have the same dimensions as 'A' and '-A'

Notes

    1. 'a' can be any number
    2. If you add any number with its opposite, you'll always get zero
    3. A number's opposite is that number multiplied by -1
    1. A matrix's opposite is a matrix that has the same dimensions, but each element is replaced by its opposite
    2. A number's opposite is that number multiplied by -1
    3. This is the same as multiplying the entire matrix by the scalar -1
    1. A zero matrix is a matrix with zeros as its elements
    2. A matrix's opposite is a matrix that has the same dimensions, but each element is replaced by its opposite
    1. Elements with the same color are in the same position in their respective matrices
    2. A matrix's opposite is a matrix that has the same dimensions, but each element is replaced by its opposite
    3. Here we just multiplied each element by -1 to get its corresponding element in the opposite matrix
    4. Adding each set of corresponding elements gives us a matrix with 0 in every position!
    1. When we added a matrix and its opposite we got a zero matrix, just like how we added a number and its opposite to get 0
    1. Adding the opposite gives us a zero matrix in our example
    2. Will adding the opposite work for any matrix, not just our example?
    3. Remember, a matrix's opposite is a matrix of the same dimensions, but with opposite elements
    1. If we can show that this works with variables, we can prove that the property holds in general!
    2. To get the opposite of this matrix, just change the sign in front of each variable in the matrix
    3. So we have -a, -b, -c, and -d as the elements in the opposite matrix
    1. Just like before, when we add each number and its opposite we get 0
    2. This gives us a matrix with all zeros as its elements, or a zero matrix
    1. Since the property worked for our general matrices, that means it holds for ALL matrices!
    1. A zero matrix is a matrix whose elements are all zeros
    2. The opposite of 'A' is 'A' multiplied by the scalar -1