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Unformatted text preview: NOTES ON SECTION 3 . 2 MA265, SECTIONS 41, 52 Key words Reduction to triangular form The formula we developed in the last section is horrendous to use. We developed it so that we would have an object called the determinant; we will now show that the determinant has nice properties which we can use in practice (instead of the definition). Theorem 1. The determinant is the unique function from square matrices to the real numbers satisfying the following 4 properties: (1) It is additive in each column (that is, if you have a column a + b , then the determinant is the sum of the determinants of the matrices you obtain by replacing that column with the columns a and b instead). (2) It is multiplicative in each column (that is, if you have a column k a , where k is a scalar and a is a column vector, then the determinant of the matrix is k times the determinant of the matrix you get by replacing the column k a with a ). (3) It is alternating, meaning that if you exchange two columns then you change the sign of the deter- minant....
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