Question

If a 2x2 matrix has the same two eigenvalues as another 2x2 matrix, what conclusions can you infer about the two matrices? Make some hypotheses and attempt to prove them, or refute them via counterexamples.

Answer 1

Answer with explanation

It is given that , two matrix, having order , 2×2, has same eigenvalue.Let P, and Q be two matrix.

Suppose,

$P=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$

→→P-αI

   Where, α is a scalar, and I is Identity matrix having order , 2×2.

 $=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]- \left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

When we evaluate the determinant , it will be in the form of Quadratic function ,which we call Characteristic Polynomial , which will give two eigen values.

→→Now, it is given that, two matrix having same order, has same eigen values , means characteristic polynomial of two equation will be same.

→→If you solve the Characteristic equation that is

| P -αI |=0

You will get two Eigenvalues.

→The two matrrix will be identical, that is same if they have same eigenvalue.