Determinants Te...

If $$\triangle_{r}=\begin{vmatrix} { 2 }^{ r-1 } & 2.{ 3 }^{ r-1 } & 4.{ 5 }^{ r-1 } \\ x & y & z \\ { 2 }^{ n-1 } & { 3 }^{ n-1 } & { 5 }^{ n-1 } \end{vmatrix}$$, then $$\sum_{r=1}^{n}(\triangle_{r})$$ is equal to

if $$a^{2},b^{2}+c^{2}+ab+bc+ca \le 0 \forall a,b,c \epsilon R$$, then value of the determinant $$\left| \begin{matrix} \left( a^{ 2 }+b^{ 2 }+c^{ 2 } \right) ^{ 2 } & a^{ 2 }+b^{ 2 } & 1 \\ 1 & (b+c+2) & b^{ 2 }+c^{ 2 } \\ c^{ 2 }+a^{ 2 } & 1 & (c+a+2)^{ 2 } \end{matrix} \right| $$ equals

Adj $$\begin{bmatrix} 1 & 0 & 2 \\ -1 & 5 & -2 \\ 0 & 2 & 1 \end{bmatrix}=\begin{bmatrix} 9 & a & -2 \\ -1 & 1 & 0 \\ -2 & 2 & b \end{bmatrix}\Rightarrow \left[ \begin{matrix} a & b \end{matrix} \right] =$$

 $$\Delta  = \left| \matrix{  1 + {a^2} + {a^4}\;\;1 + ab + {a^2}{b^2}\;\;1 + ac + {a^2}{c^2} \hfill \cr   1 + ab + {a^2}{b^2}\;\;1 + {b^2} + {b^4}\;\;1 + bc + {b^2}{c^2} \hfill \cr   1 + ac + {a^2}{c^2}\;\;1 + bc + {b^2}{c^2}\;\;1 + {c^2} + {c^4} \hfill \cr}  \right|is\;equal\;to$$                  

Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix} \, and \, U_1, U_2, U_3$$ be column matrices satisfying $$AU_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , AU_2 = \begin{bmatrix} 2 \\ 3 \\ 6 \end{bmatrix} , AU_3 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$$. If U is $$3 \times 3$$ matrix, whose columns are $$U_1, U_2, U_3$$. then |U| is 

Matrix $$A = \left| {\begin{array}{*{20}{c}}x & 3 & 2\\1 & y & 4\\2 & 2 & z\end{array}} \right|$$, if $$xyz=60$$ and $$8x+4y+3z=20$$, then $$a(adjA)$$ is equal to 

$$\begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix}=$$

The number of distinct values of a $$2 \times 2$$ determinant whose entries are from set $$\{-1, 0, 1\}$$ is

$$f\left( x \right) = \left| {\begin{array}{*{20}{c}}{x - 2}&{{{\left( {x - 1} \right)}^2}}&{{x^3}}\\{x - 1}&{{x^2}}&{{{\left( {x + 1} \right)}^3}}\\x&{{{\left( {x + 1} \right)}^2}}&{{{\left( {x + 2} \right)}^3}}\end{array}} \right|$$

If $$\theta \varepsilon R,$$ then the determinant $$\Delta =\begin{vmatrix} \sin { \theta  }  & \cos { \theta  }  & \sin { 2\theta  }  \\ \sin { \left( \theta +\dfrac { 2\pi  }{ 3 }  \right)  }  & \cos { \left( \theta +\dfrac { 2\pi  }{ 3 }  \right)  }  & \sin { \left( 2\theta +\dfrac { 4\pi  }{ 3 }  \right)  }  \\ \sin { \left( \theta -\dfrac { 2\pi  }{ 3 }  \right)  }  & \cos { \left( \theta -\dfrac { 2\pi  }{ 3 }  \right)  }  & \sin { \left( 2\theta -\dfrac { 4\pi  }{ 3 }  \right)  }  \end{vmatrix}=$$