determinant in terms of traces of powers

It is possible to express the determinant of a matrix in of traces ofpowers of a matrix.

The easiest way to derive these expressions is to specialize to the case ofdiagonal matrices. For instance, suppose we have a $2\\times 2$ matrix $M=\\operatorname{diag}(u,v)$ . Then

$\\displaystyle\\operatorname{det}M$	$\\displaystyle=$	$\\displaystyle uv$
$\\displaystyle\\operatorname{tr}M$	$\\displaystyle=$	$\\displaystyle u+v$
$\\displaystyle\\operatorname{tr}M^{2}$	$\\displaystyle=$	$\\displaystyle u^{2}+v^{2}$

From the algebraic identity $(u+v)^{2}=u^{2}+v^{2}+2uv$ , it can be concluded that $\\operatorname{det}M=\\frac{1}{2}(\\operatorname{tr}M)^{2}-\\frac{1}{2}%\\operatorname{tr}(M^{2})$ .

Likewise, one can derive expressions for the determinants of larger matrices from the identities for elementary symmetric polynomials in of power sums. For instance, from the identity

xyz=\\frac{1}{6}(x+y+z)^{3}-\\frac{1}{2}(x^{2}+y^{2}+z^{2})(x+y+z)+\\frac{1}{3}(x%^{3}+y^{3}+z^{3}),

it can be concluded that

\\operatorname{det}M=\\frac{1}{6}(\\operatorname{tr}M)^{3}-\\frac{1}{2}(%\\operatorname{tr}M^{2})(\\operatorname{tr}M)+\\frac{1}{3}\\operatorname{tr}M^{3}

for a $3\\times 3$ matrix $M$ .