SECTION 11.3 Systems of Linear Equations: Determinants 791 5 Know Properties of Determinants Determinants have several properties that are sometimes helpful for obtaining their value. We list some of them here. THEOREM The value of a determinant changes sign if any two rows (or any two columns) are interchanged. (12) Proof for 2 by 2 Determinants ( ) = − = − = − − a b c d ad bc c d a b bc ad ad bc and ■ EXAMPLE 6 Demonstrating Theorem (12) = − = = − = − 3 4 1 2 6 4 2 1 2 3 4 4 6 2 THEOREM If all the entries in any row (or any column) equal 0, the value of the determinant is 0. (13) THEOREM If any two rows (or any two columns) of a determinant have corresponding entries that are equal, the value of the determinant is 0. (14) Proof Expand across the row (or down the column) containing the 0’s. ■ In Problem 68, you are asked to prove the theorem for a 3 by 3 determinant in which the entries in column 1 equal the entries in column 3. Demonstrating Theorem (14) ( ) ( ) ( ) ( ) ( ) ( ) = − ⋅ ⋅ + − ⋅ ⋅ + − ⋅ ⋅ =−−−+−=−+ −= + + + 1 2 3 1 2 3 4 5 6 1 1 2 3 5 6 1 2 1 3 4 6 1 3 1 2 4 5 1 3 2 6 3 3 3 12 9 0 1 1 1 2 1 3 EXAMPLE 7 THEOREM If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of the determinant is also changed by a factor of k. (15) In Problem 67, you are asked to prove the theorem for a 3 by 3 determinant using row 2.
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