Solve Matrix A: Determining Trace & R(LA)

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In summary, the problem is to find a real symmetric 3x3 matrix A satisfying certain conditions such as trace(A)=0, R(LA)=span{(1,1,1),(1,0,-1)}, and A*(1,1,1)=(2,1,0). The matrix has 6 unknowns and 6 equations can be found by considering the given conditions. However, the last condition regarding R(LA) may provide two more equations, but the exact equations are unclear.
  • #1
steinmasta
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Homework Statement



Suppose A is a real symmetric 3
× 3 matrix such that
• trace(A) = 0
• R(LA) = span {(1, 1, 1) and (1, 0, -1)} (sorry for formatting issues - these are both column vectors)
where La is the left multiplication transformation

• A * (1, 1, 1) = (2, 1, 0) again, these are column vectors

Find A. Explain your answer.

Attempts at solution:
Because the matrix is symmetric, entries a12 = a21, a13=a31 and a23=a32, so there are 6 unknowns.

I need 6 equations then. The trace being zero implies a11+a22+a33 = 0, so that's one equation. Three more equations come from the last condition, multiplying A * the column vector (1, 1, 1). I need two more equations, which I think come from the condition regarding R(La), but I can't figure them out. Any help would be much appreciated.
 
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  • #2
What does R denote? Does this mean that for all x, Ax = b(1, 1, 1) + c(1, 0, -1) for some real b, c?
 

1. What is a matrix?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns.

2. What is the trace of a matrix?

The trace of a matrix is the sum of the elements on the main diagonal (top-left to bottom-right) of the matrix.

3. How do you determine the trace of a matrix?

To determine the trace of a matrix, simply add up all the elements on the main diagonal. This can be done manually or by using a calculator or computer program.

4. What is the significance of the trace of a matrix?

The trace of a matrix is often used as a measure of the matrix's similarity to another matrix, as well as in various mathematical proofs and calculations.

5. How do you find the rank of a matrix using its trace?

The rank of a matrix can be determined using its trace by comparing the trace to the number of non-zero rows or columns in the matrix. If the trace is equal to the number of non-zero rows or columns, then the rank is equal to the trace. If the trace is less than the number of non-zero rows or columns, then the rank is equal to the trace plus the number of zero rows or columns.

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