Solve {ccc}{a-b}&{b-c}&{c}{(a+b)(b-b)}&{(b+c)(b-c)}&{c^2}{-(a-b)}&{-(b-c)}&{a+b}

\left| \begin{array} { c c c } { a - b } & { b - c } & { c } \\ { ( a + b ) ( b - b ) } & { ( b + c ) ( b - c ) } & { c ^ { 2 } } \\ { - ( a - b ) } & { - ( b - c ) } & { a + b } \end{array} \right|

Evaluate

\left(a-b\right)\left(a+b+c\right)\left(b^{2}-c^{2}\right)

Integrate w.r.t. a

\frac{a\left(a^{2}-3b^{2}\right)\left(b^{2}-c^{2}\right)}{3}+\frac{ac\left(a-2b\right)\left(b^{2}-c^{2}\right)}{2}+С

Quiz

5 problems similar to:

\left| \begin{array} { c c c } { a - b } & { b - c } & { c } \\ { ( a + b ) ( b - b ) } & { ( b + c ) ( b - c ) } & { c ^ { 2 } } \\ { - ( a - b ) } & { - ( b - c ) } & { a + b } \end{array} \right|

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Prove that the paraboloids: \displaystyle\frac{{x}^{{2}}}{{{a}_{{1}}^{{2}}}}+\frac{{y}^{{2}}}{{{b}_{{1}}^{{2}}}}=\frac{{{2}{z}}}{{c}_{{1}}} ; \displaystyle\frac{{x}^{{2}}}{{{a}_{{2}}^{{2}}}}+\frac{{y}^{{2}}}{{{b}_{{2}}^{{2}}}}=\frac{{{2}{z}}}{{c}_{{2}}} ...

https://socratic.org/questions/prove-that-the-paraboloids-x-2-a-1-2-y-2-b-1-2-2z-c-1-x-2-a-2-2-y-2-b-2-2-2z-c-2-1

Cesareo R. May 12, 2017 See https://socratic.org/questions/prove-that-the-paraboloids-x-2-a-1-2-y-2-b-1-2-2z-c-1-x-2-a-2-2-y-2-b-2-2-2z-c-2

determinant linear factorisation

https://math.stackexchange.com/questions/333579/determinant-linear-factorisation

Playing around, and beginning pretty much as you began (except carrying out the row operations correctly): D = \begin{vmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c & 1 \end{vmatrix}\\ = \begin{vmatrix} (a^3-b^3)+(a^2-b^2) & a-b & 0 \\ (b^3-c^3)+(b^2-c^2) & b-c & 0 \\ c^3+c^2 & c & 1 \end{vmatrix}\\ = \begin{vmatrix} (a^3-b^3)+(a^2-b^2) & a-b \\ (b^3-c^3)+(b^2-c^2) & b-c \end{vmatrix}\\ = (a-b)(b-c) \begin{vmatrix} a^2+ab+b^2+a+b & 1 \\ b^2+bc+c^2+b+c & 1 \end{vmatrix}\\ = (a-b)(b-c)(a^2+ab+b^2+a+b-b^2-bc-c^2-b-c)\\ = (a-b)(b-c)(a^2+ab+a-bc-c^2-c)\\ = (a-b)(b-c)(a^2+ab+a+ac-ac-bc-c^2-c)\\ = (a-b)(b-c)(a-c)(a+b+c+1), ...

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https://math.stackexchange.com/questions/941616/a-determinant-problem

Let g(x) = (x-1)(x-\alpha)(x-\beta), a monic polynomial with the roots 1, \alpha, \beta. Then the Vandermonde matrix of roots to g is M = \left(\begin{array}{ccc}1 & 1 & 1\\1 & \alpha & \alpha^2\\ 1 & \beta & \beta^2\end{array}\right) ...

Determinant of a Block Matrix times Inverse

https://math.stackexchange.com/questions/992686/determinant-of-a-block-matrix-times-inverse

You can use another decomposition. \begin{pmatrix} 1 & -\alpha A^{-1} \\ 0 & A^{-1} \end{pmatrix} \begin{pmatrix} a & \alpha \\ \beta & A \end{pmatrix} = \begin{pmatrix} a - \alpha A^{-1} \beta & 0 \\ A^{-1} \beta & I \end{pmatrix} ...

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https://math.stackexchange.com/questions/755547/an-application-of-the-mountain-pass-lemma

You should set up the problem in the space W^{1,2}(]0,\pi[). Moreover the functional should be: \varphi(u):=\int_0^\pi \left(\frac12|u'|^2-\lambda u^2-\frac{u^4}{4}\right)dt which verifies the ...

5 real numbers tied by two equations: understanding a proof of the related problem.

https://math.stackexchange.com/questions/2894999/5-real-numbers-tied-by-two-equations-understanding-a-proof-of-the-related-pro

He is using it for 2ab \leq a^2+b^2 2ac \leq a^2+c^2 2ad \leq a^2+d^2 2bc \leq b^2+c^2 2bd \leq b^2+d^2 and 2cd \leq c^2+d^2

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\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]

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