For a square matrixA

, the statement

"(AI)v=0

has a non-zero solutionv

"

is equivalent to the statement

"det(AI)=0

".

The quantitydet(AI)

is a polynomial in

, known as thecharacteristic polynomialofA

, and its roots are the eigenvalues ofA

.This gives us a technique for finding the eigenvalues ofA

.

Let's check our understanding.

For a square matrix A' the statement "(A ADV : 0 has a nonzero solution v \" is equivalent to the statement "det[A 7 AI] : D The quantity det(A AI) is a polynomial in A, known as the characteristic polynomial of A, and its roots are the Eigenvalues of A, This gives us a technique for nding the eigenvalues of A, Let's check our understanding. i) The characteristic polynomial ofA : (g i) is 1: 2 -- detAtI=dt = _ ( i e(2 1_t) i In a Ordered 1 , 4 -2 0 ii) The characteristic polynomial of B = -2 4 I) is the cubic 0 0 -5 det(BtI):| inl- Hinl: try expanding the determinant along the last row or last column. Ordered 1