Determinants

Determinants - Definition of a determinant A nxn...

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1 Determinants Determinants, Cramer’s Rule Definition of a determinant A nxn determinant is a square array of numbers, constants, or variables, denoted det A or |A| computed in a particular way: it is the “sum” (with certain + and – signs) of products of n elements, one from each row and column, but never two or more from the same line or column Examples of determinants Here are two examples These letters could represent numbers, or numbers and variables, etc. Determinants and matrices With each square matrix A we can associate a eterminant det A determinant det A We call the element a ij is the ij th element of the matrix A (or determinant) Example: for the 3x3 determinant on the previous slide, h is the a 32 element
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2 Minors and cofactors For a given element a ij of a matrix (or eterminant) the inor defined as being the determinant) the minor M ij is defined as being the determinant obtained by erasing the i th row and j th column of the matrix The cofactor C ij of a ij is defined to be ) j (-1) i+j M ij Example of minors and cofactors Consider matrix (or corresponding determinant) Minor of the a 21 element is (erase second row, first column) 5x5 - 6x3 = 7 Cofactor of same element is obtained by multiplying by (-1) 2+1 = -1 Cofactor expansions (Allows expressing a determinant as a sum of aller determinants) smaller determinants) Method: Pick a row, e.g. row k Then det(A)= a k1 C k1 +a k2 C k2 +a k3 C k3 + …+ a kn C kn Example of cofactor expansion We evaluate in this way: =2(7x5 - 1x6) -5(4x5 - 3x1) + 3(4x6 - 3x7) =-18
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3 Example of cofactor expansion , cont’d Or we can expand by the second row:
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Determinants - Definition of a determinant A nxn...

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