A is a matrix of order 2x2 whose main diagonal's entries' sum is zero. Prove that A^2 is a scalar matrix?

[TheWrathOfMath]

1. A is a matrix of order 2x2 whose main diagonal's entries' sum is zero. Prove that A^2 is a scalar matrix.

2. Given: A and B are two matrices of order 2x2. Prove that the sum of the entries of the main diagonal of AB-BA is zero.

3. A, B and C are three matrices of order 2x2. Given: A^2 is a scalar matrix and the sum of the entries of the main diagonal of AB-BA is zero. Prove that C (AB-BA) ^2= (AB-BA) ^2*C?

 

[Deleted member 4993]

1. A is a matrix of order 2x2 whose main diagonal's entries' sum is zero. Prove that A^2 is a scalar matrix.

2. Given: A and B are two matrices of order 2x2. Prove that the sum of the entries of the main diagonal of AB-BA is zero.

3. A, B and C are three matrices of order 2x2. Given: A^2 is a scalar matrix and the sum of the entries of the main diagonal of AB-BA is zero. Prove that C (AB-BA) ^2= (AB-BA) ^2*C?

You ought to know the rules of this forum. One of those is that there ought to be one problem per thread. you posted 3 problems in this thread - please make separate threads for the other two problems.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[pka]

Does what is in this LINK tell you anything?

 

[Steven G]

1. A is a matrix of order 2x2 whose main diagonal's entries' sum is zero. Prove that A^2 is a scalar matrix.

2. Given: A and B are two matrices of order 2x2. Prove that the sum of the entries of the main diagonal of AB-BA is zero.

3. A, B and C are three matrices of order 2x2. Given: A^2 is a scalar matrix and the sum of the entries of the main diagonal of AB-BA is zero. Prove that C (AB-BA) ^2= (AB-BA) ^2*C?

You really have to get your hands dirty sometimes. This problem is extremely easy.
You have a 2x2 matrix which means two entreies in the 1st row and two entries in the 2nd row. The (two) diagonals add up to 0, so let the entries on the diagonal be a and -a. Choose b and c for the other two entries. Now we have the 1st row being a b and the 2nd row being c -a. Now get your hands dirty and multiply this matrix by itself. After multiplying the two matrices say wow I got a diagonal matrix!