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m235notes19 - Notes 19 Determinants Lecture November 2011...

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Notes 19: Determinants Lecture November, 2011 Let A be an n × n matrix. The determinant of A is a number denoted det ( A ) or | A | . It has many uses, but we will use it for one thing. It enables us to decide if a matrix is invertible. Theorem 1 . Let A be an n × n matrix. Then A is invertible if and only if det ( A ) = | A | 6 = 0 . Corollary 1 . Let A be an n × n matrix. The following are equivalent: det ( A ) = 0 . A is not invertible. ker ( A ) 6 = 0 . im ( A ) 6 = R n . Definition 1. Let A be an n × m. the transpose of A is the m × n matrix whose rows are the columns of A. Let A be an n × n matrix. We show how to compute the determinant of A . We do not give a formula for the determinant, but rather a set of rules. We give the value for the determinant for simple matrices and a set of rules that allow us to turn the matrix into one of the simple matrices while keeping track of how the value of the determinant changes. Rule 1: The determinant of a matrix with all zeros above the diagonal is the product of the diagonal entries. The determinant of a matrix with all zeros below the diagonal is
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