NxN matrix multiplication; commutativity

In summary, matrix multiplication is not commutative for all N > 1, as shown by the examples of multiplying 2x2 and 3x3 matrices. This can be explicitly demonstrated by defining matrices with mostly zeros and a few non-zero entries.
  • #1
207
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Homework Statement


Explicitly show, using numbers, that in general NxN matrix multiplication is not commutative. Show this for all N > 1.


If I make a matrix A, 2x2,

1 2
3 4

and multiply by matrix B, also 2x2

5 6
7 8

I get AB =/=BA

If I multiply a new matrix A, 3x3,

1 2 3
4 5 6
7 8 9

by a new matrix B, 3x3,

10 11 12
13 14 15
16 17 18

I get AB=/=BA

Ok... but how do I show that this holds for ALL n > 1? I have to do this explicitly using real numbers.
 
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  • #2
There is a simple way to make a smaller matrix into a larger matrix.
 
  • #3
hi mathman44! :smile:
mathman44 said:
Ok... but how do I show that this holds for ALL n > 1? I have to do this explicitly using real numbers.

try defining a matrix that's nearly all zeros, and a few 1s or -1s :wink:
 

1. What is an NxN matrix?

An NxN matrix is a matrix with equal number of rows and columns. The 'N' in NxN represents the size of the matrix, which can vary depending on the context.

2. What is matrix multiplication?

Matrix multiplication is a mathematical operation where two matrices are multiplied together to produce a third matrix. It involves multiplying each element of one matrix with the corresponding element of the other matrix and adding them to get the element of the resulting matrix.

3. What is meant by commutativity in matrix multiplication?

Commutativity in matrix multiplication means that the order of the matrices being multiplied does not affect the result. This means that the result of multiplying matrix A by matrix B will be the same as multiplying matrix B by matrix A.

4. Is matrix multiplication commutative for all matrices?

No, matrix multiplication is not commutative for all matrices. It is commutative only for square matrices, i.e. matrices with equal number of rows and columns. For non-square matrices, the order of multiplication will affect the result.

5. What is the significance of commutativity in matrix multiplication?

Commutativity in matrix multiplication allows for more efficient calculation and manipulation of matrices. It also helps in simplifying equations and finding solutions to systems of linear equations. Additionally, it is a fundamental property in many mathematical and scientific fields, such as physics and engineering.

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