If you have a square matrix (for example A=
1 2 2
2 1 2
2 2 1
and you want to show that this matrix is a zero of the polynomial P(x)=
x²-4x-5,
can you also do this by calculating the determinant of the matrix A and filling this number into the polynomial function?
Or is that just a coincidence in this case?
Answer
First, when we say that matrix A satisfies x^2 -4x – 5, we mean that
AA – 4A – 5 In = 0n
where the unit matrix In is and 0n the zero matrix (so not the number zero)
The fact that the determinant also satisfies this is a coincidence.
I give two counterexamples where it is not the case.
1) Take for example the 2×2 matrix A = [2 0; 0 2]so the double of the unit matrix
it satisfies the simple equation x – 2 = 0
However, its determinant is 4, and does not satisfy this equation
2) Conversely, this determinant of course satisfies the equation x – 4 = 0
but then again the matrix does not comply.
Now it is true that a square matrix always satisfies its own characteristic equation. That’s the Cayley-Hamilton theorem. You may have heard about that in connection with matrices and determinants. After all, the characteristic equation p(λ)= 0 is : determinant (A – λ . I) = 0
Answered by
prof.dr. Paul Hellings
Department of Mathematics, Fac. IIW, KU Leuven
Old Market 13 3000 Leuven
https://www.kuleuven.be/
.