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 I_{n} = 0n_{}

where the unit matrix I_{n } is and 0_{n} 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/

.