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{\displaystyle C_{n}={\begin{bmatrix}a_{1n}\\a_{2n}\\a_{3n}\\\vdots \\a_{mn}\\\end{bmatrix}}}
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A linear combination of columns n1 C1 +n2 C2 +n3 C3 +...+nn Cn is the column
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{\displaystyle C_{n}={\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\\\vdots \\c_{n}\\\end{bmatrix}}}
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Where ck =n1 ak1 +n1 ak1 +n2 ak2 +n3 ak3 +...+nn akn .
If there is a determinant of order n which is A=aij , and there are n columns of n elements such that the ith entry of the jth column is equal to aij , then if one of the columns is a linear combination of the other columns, then the determinant is equal to 0.
Suppose that the kth column is a linear combination of the other column,
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{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}&\ldots &c_{1}a_{11}+c_{2}a_{12}+c_{3}a_{13}+\ldots +c_{n}a_{1}n&\ldots &a_{1n}\\a_{21}&a_{22}&a_{23}&\ldots &c_{1}a_{21}+c_{2}a_{22}+c_{3}a_{23}+\ldots +c_{n}a_{2}n&\ldots &a_{2n}\\a_{31}&a_{23}&a_{33}&\ldots &c_{1}a_{31}+c_{2}a_{32}+c_{3}a_{33}+\ldots +c_{n}a_{3}n&\ldots &a_{3n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n3}&a_{n3}&\ldots &c_{1}a_{n1}+c_{2}a_{n2}+c_{3}a_{n3}+\ldots +c_{n}a_{n}n&\ldots &a_{nn}\\\end{bmatrix}}}
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Then by the linearity of determinants, the determinant is equal to
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{\displaystyle c_{1}{\begin{bmatrix}a_{11}&a_{12}&a_{13}&\ldots &a_{11}&\ldots &a_{1n}\\a_{21}&a_{22}&a_{23}&\ldots &a_{21}&\ldots &a_{2n}\\a_{31}&a_{23}&a_{33}&\ldots &a_{31}&\ldots &a_{3n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n3}&a_{n3}&\ldots &a_{n1}&\ldots &a_{nn}\\\end{bmatrix}}+c_{2}{\begin{bmatrix}a_{11}&a_{12}&a_{13}&\ldots &a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&a_{23}&\ldots &a_{22}&\ldots &a_{2n}\\a_{31}&a_{23}&a_{33}&\ldots &a_{32}&\ldots &a_{3n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n3}&a_{n3}&\ldots &a_{n2}&\ldots &a_{nn}\\\end{bmatrix}}+c_{3}{\begin{bmatrix}a_{11}&a_{12}&a_{13}&\ldots &a_{13}&\ldots &a_{1n}\\a_{21}&a_{22}&a_{23}&\ldots &a_{23}&\ldots &a_{2n}\\a_{31}&a_{23}&a_{33}&\ldots &a_{33}&\ldots &a_{3n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n3}&a_{n3}&\ldots &a_{n3}&\ldots &a_{nn}\\\end{bmatrix}}+\ldots +c_{n}{\begin{bmatrix}a_{11}&a_{12}&a_{13}&\ldots &a_{1n}&\ldots &a_{1n}\\a_{21}&a_{22}&a_{23}&\ldots &a_{2n}&\ldots &a_{2n}\\a_{31}&a_{23}&a_{33}&\ldots &a_{3n}&\ldots &a_{3n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n3}&a_{n3}&\ldots &a_{nn}&\ldots &a_{nn}\\\end{bmatrix}}}
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Since all of those matrices have repeat columns, their determinants are 0, and so their sum is 0.
The rank of a matrix is the maximum order of a minor that does not equal 0. The minor of a matrix with the order of the rank of the matrix is called a basis minor of the matrix, and the columns that the minor includes are called the basis columns.