Algebra 2 Notes 3 Flashcards
(3D-N1-3.1) Determinant of a matrix
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(3P-N2-3.2) Given 2x2 matrix. (i) det(A) = ad − bc ii) A is invertible iff det(A) =/ 0. In this case inverse A = … (iii) Let LA be the linear map from R2 to R2 given by LA(v) = Av. Then if S is a shape in R2, then Area(LA(S)) = | det A|× Area(S). (iv) If B is another 2 × 2 matrix, then det(AB) = det A det B.
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(3P-N2-3.3) For detA = a11a22a33 + a12a23a31 + a13a21a32 − a12a21a33 − a13a22a31 − a11 a23 a32
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(3P-N3-3.4) Let A be an n × n matrix: then det(AT ) = det(A).
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(3P-N3-3.5) Let A be a lower triangular matrix, i.e. one such that aij = 0 for all j > i. Then det(A) = a11a22…ann.
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(3T-N4-3.6) (a) Exchanging two rows of a matrix multiplies the determi- nant by -1.(b) Multiplying a row of a matrix by λ multiplies the determinant by λ. (c) Adding a multiple of one row to another row leaves the determinant unchanged.
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(3C-N5-3.7) Let A be a square matrix and E1,…,En elementary matrices of the same size. Then det(En…E2E1A) = det(En)… det(E2) det(E1) det(A).
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(3T-N5-3.8) Let A be an m × m matrix. A is invertible ⇔ det(A) ̸= 0.
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(3T-N6-3.10) Let A, B be m×m matrices. Then det(AB) = det(A) det(B)
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(3D-N6-3.11) The (i, j)-minor Mij of an n × n matrix A is the determinant of the n − 1 × n − 1 matrix obtained by deleting the ith row and jth column of A, i.e. removing the row and column in which aij occurs The (i,j)-cofactor of A is Cij = (−1)i+jMij.
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(3P-N6-3.12) Let A be an n×n matrix. Then for any i, det(A)
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(3D-N7-3.13) Adjugate of A,
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(3T-N7-3.14) A(adj(A)) = adj(A)A = det(A)In. Hence if A is invertible, its inverse is given by
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(3D-N8-E1) The following elementary row operations can be carried out on matrices:
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(3D-N8-E2) Corresponding to each elementary row operation e
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