Theorem 3. Let A be any n x n matrix. (a) If A' is the resulting matrix when the single row A is multiplied by the constant k, then det(A) = k sec(A). (b) If A' is the resulting matrix when two rows of A are interchanged, then det(A')=-det(A). (c) If A' is the resulting matrix when a multiple of one row A is added to another row, then det(A')=det(A).

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Prove the following theorem. An example of proof can be seen in the picture, maybe it can help you

Theorem 3. Let A be any n x n matrix.

(a) If A' is the resulting matrix when the single row A is multiplied by the constant k, then det(A) = k sec(A).

(b) If A' is the resulting matrix when two rows of A are interchanged, then det(A')=-det(A).

(c) If A' is the resulting matrix when a multiple of one row A is added to another row, then det(A')=det(A).