<s>[INST] Finding determinant using properties of determinant without expanding show that determinant
$$\left|\matrix{
x^2+L  &  xy   &   xz \\
xy   &    y^2+L &   yz \\
xz   &    yz   &   z^2+L \\
}\right|
=  L^2(x^2+y^2+z^2+L)$$
without expanding by using the appropriate properties of determinant.
All i can do is LHS
$$x^2y^2z^2\left|\matrix{
1+L/x^2  & 1  &   1 \\
1   &   1+L/y^2 &   1 \\
1  &    1   &   1+L/z^2 \\
}\right|$$
Is it a must to relate to eigenvalue problem? 
 [/INST] $$\left|\matrix{
x^2+L  &  xy   &   xz \\
xy   &    y^2+L &   yz \\
xz   &    yz   &   z^2+L \\
}\right|
= $$
$$xyz\left|\matrix{
x+L/x  &  y   &   z \\
x   &    y+L/y &   z \\
x   &    y   &   z+L/z \\
}\right|
= $$
$$=\left|\matrix{
x^2+L  &  y^2   &   z^2 \\
x^2   &    y^2+L &   z^2 \\
x^2   &    y^2   &   z^2+L \\
}\right|
= $$
$$=\left|\matrix{
x^2+y^2+z^2+L  &  y^2   &   z^2 \\
x^2+y^2+z^2+L   &    y^2+L &   z^2 \\
x^2+y^2+z^2+L   &    y^2   &   z^2+L \\
}\right|
= $$ $$=(x^2+y^2+z^2+L)\left|\matrix{
1  &  y^2   &   z^2 \\
0   &    L &    0\\
0   &    0   &   L \\
}\right|=L^2(x^2+y^2+z^2+L)$$
</s>