2.3 - MATH1850U/2050U: Chapter 2 cont. 1 DETERMINANTS cont...

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MATH1850U/2050U: Chapter 2 cont. .. 1 DETERMINANTS cont… Properties of Determinants; Cramer’s Rule (2.3; pg. 106) Theorem: For an n n matrix A , ) det( ) det( A k kA n where k is any scalar. Example: Find ) 3 det( A if 2 ) det( A where A is 4 4 . Caution: In general, ) det( ) det( ) det( B A B A Example: Verify that ) det( ) det( ) det( B A B A for the given matrices. Question: Is there a rule for the determinant of a product of two matrices? Let’s start with the special case when one of the matrices is an elementary matrix. Lemma: If B is an n n matrix and E is an n n elementary matrix, then ) det( ) det( ) det( B E EB Theorem: A square matrix is invertible if and only if 0 ) det( A . Proof:
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MATH1850U/2050U: Chapter 2 cont. .. 2 Theorem: If A and B are square matrices of the same size, then ) det( ) det( ) det( B A AB That is, the determinant of the product is equal to the product of the determinants. Example:
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2.3 - MATH1850U/2050U: Chapter 2 cont. 1 DETERMINANTS cont...

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