关于高等代数的计算题II(行列式)

$f计算：$$f(02浙大二)$设${S_n} = {x_1}^k + {x_2}^k +  cdots  + {x_n}^kleft( {k = 0,1,2, cdots } ight),{a_{ij}} = {S_{i + j - 2}}left( {i,j = 1,2, cdots ,n} ight)$，计算[D = left| {egin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}& cdots &{{a_{1n}}} \ 
{{a_{21}}}&{{a_{22}}}& cdots &{{a_{2n}}} \ 
cdots & cdots & cdots & cdots \ 
{{a_{n1}}}&{{a_{n2}}}& cdots &{{a_{nn}}} 
end{array}} ight|]

1

$f计算：$$f(11华师一)$

[{D_n} = left| {egin{array}{*{20}{c}}
{2 + {x_1}}&{2 + x_1^2}& cdots &{2 + x_1^n} \
{2 + {x_2}}&{2 + x_2^2}& cdots &{2 + x_2^n} \
vdots & vdots & vdots & vdots \
{2 + {x_n}}&{2 + x_n^2}& cdots &{2 + x_n^n}
end{array}} ight|]

1

$f计算：$$f(07武大二)$

[{D_n} = left| {egin{array}{*{20}{c}}
{{a_1} + b}&{{a_2}}& cdots &{{a_n}} \
{{a_1}}&{{a_2} + b}& cdots &{{a_n}} \
vdots & vdots & ddots & vdots \
{{a_1}}&{{a_2}}& cdots &{{a_n} + b}
end{array}} ight|]

1

$f计算：$$f(11华科一)$

[{D_n} = left| {egin{array}{*{20}{c}}
1&{{a_1}}& cdots &{{a_{n - 1}}} \
{{b_1}}&{{x_1}}& cdots &0 \
cdots & cdots & cdots & cdots \
{{b_{n - 1}}}&0& cdots &{{x_{n - 1}}}
end{array}} ight|]

1

$f计算：$$f(09浙大二)$

[{D_n} = left| {egin{array}{*{20}{c}}
{2{a_1}{b_1}}&{{a_1}{b_2} + {a_2}{b_1}}& cdots &{{a_1}{b_n} + {a_n}{b_1}} \
{{a_2}{b_1} + {a_1}{b_2}}&{2{a_2}{b_2}}& cdots &{{a_2}{b_n} + {a_n}{b_2}} \
vdots & vdots & ddots & vdots \
{{a_n}{b_1} + {a_1}{b_n}}&{{a_n}{b_2} + {a_2}{b_n}}& cdots &{2{a_n}{b_n}}
end{array}} ight|]

1

$f计算：$$f(12中科院三)$

[{D_n} = left| {egin{array}{*{20}{c}}
{{a_1}^2}&{{a_1}{a_2} + 1}& cdots &{{a_1}{a_n} + 1} \
{{a_2}{a_1} + 1}&{{a_2}^2}& cdots &{{a_2}{a_n} + 1} \
cdots & cdots & ddots & cdots \
{{a_n}{a_1} + 1}&{{a_n}{a_2} + 1}& cdots &{{a_n}^2}
end{array}} ight|]

1

$f计算：$$f(10中科院一)$

[{D_n} = left| {egin{array}{*{20}{c}}
{1 + {a_1} + {x_1}}&{{a_1} + {x_2}}& cdots &{{a_1} + {x_n}} \
{{a_2} + {x_1}}&{1 + {a_2} + {x_2}}&{}&{{a_2} + {x_n}} \
vdots & vdots & ddots & vdots \
{{a_n} + {x_1}}&{{a_n} + {x_2}}&{}&{1 + {a_n} + {x_n}}
end{array}} ight|]

1

$f计算：$$f(04南开一)$设$n$阶反对称阵$A=(a_{ij})$的行列式为1，对任意的$x$，计算$B=(a_{ij}+x)$的行列式

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$f计算：$