Properties of Determinants. In this problem you are asked to prove some of the properties of determinants that will be useful when we calculate 3 x 3 or 4 × 4 determinants. These properties are true for matrices of any dimension, but for simplicity in this problem I would like you to prove them for a general 2 × 2 matrix b] A = [cd] Prove each of the following statements: a) The interchange of rows and columns does not change the value of the deter- minant, i.e. |A| = |A'|. (Property I) b) The interchange of two rows will change the sign but not the numerical value of the determinant. (Property II) c) The multiplication of any row by a scalar k will change the value of the determinant by k-fold. (Property III)

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