Most of the diagonalization i encounter in my exercises are $4\times 4$ matrices and when I compute the determinant with $\lambda$, a lot of times I end up with a cubic or quartic polynomial.
It makes it hard for me to solve lambda to get eigenvalues after that, is there a way to avoid it while calculating the determinant? If not, how am i suppose to find the 'zero' solution for $\lambda$ when i have cubic or quartic polynomial. Thanks!
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You can often get pretty far by remembering that the sum of the eigenvalues is the trace and their product is the determinant. Unless your professor is a sadist, he will often ask you to figure our the eigenvalues of $4 \times 4$ matrices that are singular. In that case, you know that at least one eigenvalue is zero. If the matrix has rank 2, you're even luckier. The eigenvectors are the linear combinations of the columns that produce the zero vector. Then you can guess the last two from the trace.
You can't avoid it in general.
Here is a formula to solve cubic polynomial and here is a formula to solve quartic formula.
However, sometimes, exam questions are designed such that observations is useful in factorizing such polynomials. If an exam question is testing you concepts of characteristic polynomial, the challenge should not be in using those complicated formula. In particular, usually rational root test is more helpful in examination setting.