Why is matrix addition undefined when matrices have different dimensions?
Conditions
- Matrices have different dimensions (e.g., one is 3x2 and the other is 2x2).
Reasoning, step by step
- Attempt to pair entries at the same index .
- Identify an index present in one matrix but absent in the other.
- Conclude that the standard entrywise rule cannot be applied consistently.
- State that the operation is undefined under standard definitions.
Example
The instructor constructs a new problem at the bottom of the screen, writing out a 3x2 matrix and proposing to add it to a 2x2 matrix. He pauses to ask how this specific expression should be defined. Standard matrix addition and subtraction require matching row and column counts. These matrices cannot match every position one-to-one, so the standard operation is undefined.
Common misconceptions
- Believing matrices of any size can be added or subtracted.
- Assuming that some implicit padding or truncation makes the operation valid.
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