{"id":5224,"date":"2014-04-20T06:05:56","date_gmt":"2014-04-19T21:05:56","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5224"},"modified":"2014-08-01T18:34:17","modified_gmt":"2014-08-01T09:34:17","slug":"theorems-on-determinants","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5224","title":{"rendered":"Theorems on determinants"},"content":{"rendered":"
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  1. The value of a determinant remains the same if rows and columns are interchanged. In symbols, \\det(A) = \\det(A^T). <\/li>\n
  2. If all elements of any row [or column] are zero except for one element, then the value of the determinant is equal to the product of that element by its cofactor. In particular, if all elements of a row [or column] are zero the determinant is zero. <\/li>\n
  3. An interchange of any two rows [or columns] changes the sign of the determinant. <\/li>\n
  4. If all elements in any row [or column] are multiplied by a number, the determinant is also multiplied by this number. <\/li>\n
  5. If any two rows [or columns] are the same or proportional, the determinant is zero. <\/li>\n
  6. If we express the elements of each row [or column] as the sum of two terms, then the determinant can be expressed as the sum of two determinants having the same order. <\/li>\n
  7. If we multiply the elements of any row [or column] by a given number and add to corresponding elements of any other row [or column], then the value of the determinant remains the same. <\/li>\n
  8. If A and B are square matrices of the same order, then
    \n\\det(AB) = \\det(A)\\det(B)\\cdots(11)<\/li>\n
  9. The sum of the products of the elements of any row [or column] by the cofactors of another row [or column] is zero. In symbols,
    \n\\displaystyle \\sum^n_{k=1}a_{qk}A_{pk} = 0 or \\displaystyle \\sum^n_{k=1}a_{kq}A_{kp} = 0 if p \\ne q\\cdots(12)<\/p>\n

    If p = q , the sum is \\det(A) by (10). <\/p>\n<\/li>\n

  10. Let v_1,\\ v_2,\\ \\cdots,\\ v_n represent row vectors [or column vectors] of a square matrix A of order n<\/em>. Then \\det(A) = 0 if and only if there exist constants [scalars] \\lambda_1,\\ \\lambda_2,\\ \\cdots,\\ \\lambda_n not all zero such that
    \n\\lambda_1v_1 + \\lambda_2v_2 + \\cdots + \\lambda_nv_n = O \\cdots(13)<\/p>\n

    where O<\/em> is the null or zero row matrix. If condition (13) is satisfied we say that the vectors v_1,\\ v_2,\\ \\cdots,\\ v_n are linearly dependent<\/em>. A matrix A such that \\det(A) = 0 is called a singular matrix<\/em>. If \\det(A) \\ne 0, then A is a non-singular matrix<\/em>. <\/p>\n<\/li>\n<\/ol>\n

    In practice we evaluate a determinant of order n<\/em> by using Theorem 7 successively to replace all but one of the elements in a row or column by zeros and then using Theorem 2 to obtain a new determinant of order n<\/em> – 1. We continue in this manner, arriving ultimately at determinants of order 2 or 3 which are easily evaluated. <\/p>\n<\/blockquote>\n