Thread: Prove that LHS=RHS using ONLY properties of Det.

1. Prove that LHS=RHS using ONLY properties of Det.

Q) Without directly expanding the det., but using only the well-known properties, prove that:

__________________________________________________ ________

Tried solving many times but couldn't find a proper method/approach. Keep getting stuck every time...

2. Let A denote the matrix $\begin{bmatrix}-bc&b^2+bc&c^2+bc\\ a^2+ac&-ac&c^2+ac\\ a^2+ab&b^2+ab&-ab\end{bmatrix}$. The eigenvalues of A are the roots of the equation $\begin{vmatrix}-bc-\lambda&b^2+bc&c^2+bc\\ a^2+ac&-ac-\lambda&c^2+ac\\ a^2+ab&b^2+ab&-ab-\lambda\end{vmatrix} = 0$. If we try putting $\lambda = -(ab+bc+ca)$, then that equation becomes $\begin{vmatrix}a(b+c)&b(b+c)&c(b+c)\\ a(a+c)&b(a+c)&c(a+c)\\ a(a+b)&b(a+b)&c(a+b)\end{vmatrix} = 0$. The vector (x,y,z) will be an eigenvector for $\lambda$ if $\begin{bmatrix}a(b+c)&b(b+c)&c(b+c)\\ a(a+c)&b(a+c)&c(a+c)\\ a(a+b)&b(a+b)&c(a+b)\end{bmatrix}\begin{bmatrix}x\ \y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$. But that system of equations reduces to $ax+by+cz=0$, which has two linearly independent solutions. Therefore $\lambda = -(ab+bc+ca)$ is indeed an eigenvalue of A, with multiplicity at least 2. But the sum of the three eigenvalues of A is equal to the trace of A, which is $-(ab+bc+ca)$. Therefore the third eigenvalue must be $ab+bc+ca$. Finally, the determinant of A is the product of its eigenvalues, namely $(ab+bc+ca)^3$.

I don't know if that counts as a proof "using ONLY properties of Det.", but at least it didn't involve expanding the determinant.

3. Thanks for reply Opalg, much appreciated!... Using eigenvalues will work to. Main thing, was not to use expansion.

BTW "properties of determinants" mean manipulation of the det. by basic math functions (+-/*), changing of rows/columns/etc...

Yeah! got it, after a fresh start...

$
L.H.S.=
\begin{bmatrix}
-bc&b^2+bc&c^2+bc\\
a^2+ac&-ac&c^2+ac\\
a^2+ab&b^2+ab&-ab\\
\end{bmatrix}
$

$
=1/abc
\begin{bmatrix}
-abc&ab^2+abc&ac^2+abc\\
a^2b+abc&-abc&bc^2+abc\\
a^2c+abc&b^2c+abc&-abc\\
\end{bmatrix}
$

$
=abc/abc
\begin{bmatrix}
-bc&ab+ac&ac+ab\\
ab+bc&-ac&bc+ab\\
ac+bc&bc+ac&-ab\\
\end{bmatrix}
$

$
=
\begin{bmatrix}
-bc+ab+bc+ac+bc&ab+ac-ac+bc+ac&ac+ab+bc+ab-ab\\
ab+bc&-ac&bc+ab\\
ac+bc&bc+ac&-ab\\
\end{bmatrix}
$

$
=
\begin{bmatrix}
ab+bc+ca&ab+bc+ca&ab+bc+ca\\
ab+bc&-ac&bc+ab\\
ac+bc&bc+ac&-ab\\
\end{bmatrix}
$

$
=
(ab+bc+ca)
\begin{bmatrix}
1&1&1\\
ab+bc&-ac&bc+ab\\
ac+bc&bc+ac&-ab\\
\end{bmatrix}
$

$
=
(ab+bc+ca)
\begin{bmatrix}
0&1&1\\
0&-ac&bc+ab\\
ac+bc+ca&bc+ac&-ab\\
\end{bmatrix}
$

$
=
(ab+bc+ca)^2
\begin{bmatrix}
1&1\\
-ac&bc+ab\\
\end{bmatrix}
$

$
=
(ab+bc+ca)^2
\begin{bmatrix}
1&0\\
-ac&ab+bc+ca\\
\end{bmatrix}
$

$
=
(ab+bc+ca)^3
$

$
=R.H.S. [Proved]
$

4. Originally Posted by romit
Yeah! got it, after a fresh start...
Nice proof!