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Condensing variables of a matrix

I'm working with matrices such as

M = matrix([
    [a, b, 0], 
    [c, 0, d], 
    [0, e, 0]])

where a, b, c, d, and e are variables. Then I loop through some numbers for each of a, b, c, d, and e, looking at the eigenvalues of the resulting matrix. $$ char(M) = x^3 - ax^2 - (bc + de)x + ade $$ So, I'm doing more work than necessary, and ideally, I'd reduce M down to

 M = matrix([
    [a, b, 0], 
    [1, 0, d], 
    [0, 1, 0]])

Is there a simple way of using Sage to do so?