Finding Eigenvalues of a 3x3 Matrix INVOLVING LAMBDA

Question

Need some help with determinants involving eigen's.

I understand the steps used in the process below, but I don't understand how my teacher knew that he had to do those steps to get a nice 0 row with a simple determinant. He has done this multiple times in the course and just assumed we know how to do this, and I've never learned it.

Note: I checked around 5 questions with similar titles but none deal with this exactly.

EDIT: Also how come you can add columns?

Answer 1

He’s using row and column reduction to find the determinant of a matrix. This technique relies on several properties of determinants: exchanging two rows/columns changes the sign of the determinant; multiplying a row/column by a scalar multiplies the determinant by the same amount; and adding a multiple of a row/column to another doesn’t affect the value of the determinant. See this answer for an extended example.

Answer 2

There is no algorithm like RREF to clean up a matrix of polynomials whose determinant is the characteristic polynomial. But there is no harm in trying; you can try to make some entry zero or small, and in case the matrix was constructed so as to be diagonalised easily, this often works (moreover it often gives you the characteristic polynomial in (partially) factored form. And if it does not work, one can always revert to computing the determinant by expansion.