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Unformatted text preview: University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23 Winter 2008
Sophie Chrysostomou Cramer’s Rule
theorem 0.1 If Ax = b is a system of n linear equations in n unknowns
and det(A) = 0, then the unique solution x = [x1 , x2 , · · · , xn ] is of the form
xk = det(Bk )
det(A) for k = 1, · · · n where Bk is the matrix A with the k th column replaced by the column vector
example 0.2 Use Cramer’s Rule to solve the system:
x1 + x2 + x3 = 0
2x1 − x2
x2 + 4x3 = 3 1 definition 0.3 Let A be an n × n matrix.
1. Let Aij denote the (n − 1) × (n − 1) matrix obtained by removing the
ith row and j th column of A.
2. The cofactor of aij of A is
aij = (−1)i+j det(Aij ) = (−1)i+j |Aij |
3. Let A = [ aij ] be the matrix with ij th entry the ij th cofactor of A.
Then the adjoint of A denoted by adj (A) is the n × n matrix
adj (A) = (A )T
theorem 0.4 If A is an n × n matrix then
(adj (A)) A = A (adj (A)) = det(A) 2 corollary 0.5 If A is an n × n matrix and det(A) = 0, then
A− 1 = 1
det(A) 3 ...
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