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**Unformatted text preview: **Math 313 Lecture #9 2.1: The Determinant of a Matrix Another Way to Determine Invertibility. Recall that an n n matrix is invertible if and only if A is row equivalent to I . Thus, one way to determine the invertibility of A is to row reduce it. There is another way to determine if A is invertible: it is possible to attach to A a scalar, det( A ), whose value determines whether A is invertible or not. We will start with the simplest square matrices and construct this scalar for larger square matrices. Case 1: 1 1 Matrices . If A = [ a ] is a 1 1 matrix, then A is invertible if and only if a 6 = 0, the inverse being A- 1 = [1 /a ]. Define the determinant of a 1 1 matrix A = [ a ] by det( A ) = a . Case 2: 2 2 Matrices . Let A = a 11 a 12 a 21 a 22 . Suppose that a 11 6 = 0. Then row reduction gives a 11 a 12 a 21 a 22 R 2- ( a 21 /a 11 ) R 1 R 2 a 11 a 12 a 22- a 12 a 21 /a 11 a 11 R 2 R 2 a 11 a 22 a 11 a 22- a 12 a 21 . Thus when a 11 6 = 0, the matrix A is row equivalent to I if and only if a 11 a 22- a 12 a 21 6 = 0, i.e., the determinant of a 1 1 submatrix is nonzero. Now, suppose that a 11 = 0. Then switching the rows of A gives a 21 a 22 a 12 which is row equivalent to I if and only if a 21 a 12 6 = 0 (neither entry on the diagonal can be zero). But a 21 a 12 6 = 0 is the same as a 11 a 22- a 12 a 21 6 = 0 when a 11 = 0, and so we arrive at the same expression as when a 11 6 = 0....

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