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المرحلة 2
أستاذ المادة عقيل كتاب مزعل الخفاجي
4/19/2011 7:34:58 AM
Now we’ll give a first motivation of the determinant.
Theorem 1.1 (Criteria of invertibility). A square matrix A is invertible if and only if
detA 6= 0.
Proof. Let’s use elementary row operations to transform a matrix A to its triangular (rowechelon)
form. Let’s note, that if the determinant was not equal to 0, then it will not be
equal to 0 after elementary row operations, and if it was equal to 0, it will be equal to 0 after
elementary row operations. The matrix A is invertible, if it’s REF doesn’t have a row of zeros,
i.e. the determinant of its REF is not equal to 0. So, the determinant of the initial matrix
is not equal to 0. Moreover, A is not invertible if we have a row of zeros in its REF, so the
determinant of REF equals to 0, and so, the determinant of the initial matrix A is equal to
0.This theorem gives us a criteria, when the matrix is invertible. But if we know that the
determinant is not equal to 0, and so the matrix is invertible, the theorem doesn’t give us a
method of computing the inverse.
Now let’s continue with properties of determinants.This theorem gives us a criteria, when the matrix is invertible. But if we know that the
determinant is not equal to 0, and so the matrix is invertible, the theorem doesn’t give us a
method of computing the inverse.
Now let’s continue with properties of determinants.
Theorem 1.3 (The determinant of the transpose). detA> = detA.Proof. The determinant of A> is equal to the sum of all possible products of matrix elements,
taken 1 from each column and 1 from each row, as well as the determinant of A. So we have
to check that the products are included with the same signs.
To figure out the sign before a1k1a2k2 : : : a3k3 in the expression for the determinant of A>, we
have to reorder multiplicands by the second subscript. So, we can interchange multiplicands,
and transpositions will occur simultaneously in the first subscripts and in the second subscripts.
So, the sign of the final permutation will be the same, i.e. if at the end we’ll get al11al22 : : : alnn,
then
sgn(k1; k2; : : : ; kn) = sgn(l1; l2; : : : ; ln);
and so, detA = detA>.
From this theorem it follows that all properties which hold for rows of the matrix, hold for
its columns, i.e.
² Interchanging of columns changes the sign of the determinant;
² If two columns are equal, the determinant is equal to 0;
² If there is a column of 0’s, then the determinant is equal to 0.
² If we multiply a column by c, the determinant is multiplied by c as well;
² If we add r’s column multiplied by c to the s’s column, the determinant will not change.
Theorem 1.5 (Determinant of the product). For any square matrices A and B
det(AB) = detAdetB.
Remark 1.6. For the addition this fact is not true:
det(A + B) 6= detA + detB:
The proof of this theorem is quite complicated, and you will be able to find it in the
addendum to this lecture. It is too theoretical, and requires additional general theorem which
we will not state here, but it will be stated and proved in addendum.
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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