31.3. Invariant Properties under Similarity

Try these first

1. 1.

   What property do similar matrices share regarding their determinants?

   Hint

   Think about the scaling factor of transformations.

2. 2.

   Is the trace of similar matrices equal?

   Hint

   Consider what trace represents in the context of matrices.

3. 3.

   What invariant property do similar matrices share?

   • Determinant
   • Only the eigenvector
   • Trace only

   Hint

   Think about the properties that change when matrices are transformed.

4. 4.

   True or False: The trace of similar matrices can differ.

   • True
   • False

   Hint

   Recall what the trace represents.

5. 5.

   Given two matrices A and B with determinants of 5 and 5 respectively, determine if they are similar if both have the same rank of 2.

   Hint

   Consider all five invariant properties.

6. 6.

   Create an example of two matrices (A and B) that are similar. Demonstrate and calculate at least three invariant properties.

   Hint

   Use the similarity condition B = P⁻¹AP and verify the properties.