The area of the parallelogram determined by u and v A u v O u v det u v det u v

The area of the parallelogram determined by u and v a

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The area of the parallelogram determined by u and v : A u v = O u v · det u v = det u v Determinants of Order n For A M n × n ( F ) , for n 2 , denote the ( n - 1) × ( n - 1) matrix obtained from A by deleting row i and column j by ˜ A ij . Definition Let A M n × n ( F ) . If n = 1 , define det( A ) = A 11 . For n 2 , define det( A ) = n j =1 ( - 1) 1+ j A 1 j · det( ˜ A 1 j ) , where det( A ) or | A | is the determinant of A and c ij = ( - 1) i + j det( ˜ A ij ) is the cofactor of A ij . Note that det( A ) = A 11 c 11 + A 12 c 12 + · · · + A 1 n c 1 n , the cofactor expansion along the first row of A . Linearity Theorem 4.3 The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed: det a 1 . . . a r - 1 u + kv a r +1 . . . a n = det a 1 . . . a r - 1 u a r +1 . . . a n + k det a 1 . . . a r - 1 v a r +1 . . . a n for k scalar and u, v, a i row vectors in F n .
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