Continuation of Exploration 2 Let be an matrix. (a) Prove that the determinant of A

Chapter 7, Problem 7.1.1.158

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Continuation of Exploration 2 Let \(A=\left[a_{i j}\right]) be an \(n \times n\) matrix.

\((a) Prove that the determinant of A changes sign if two rows or two columns are interchanged. Start with \(3 \times 3\) a matrix and compare the expansion by expanding by the same row (or column) before and after the interchange. [Hint: Compare without expanding the minors.] How can you generalize from the \(3 \times 3\) case?

(b) Prove that the determinant of a square matrix with two identical rows or two identical columns is zero.

(c) Prove that if a scalar multiple of a row (or column) is added to another row (or column) the value of the determinant of a square matrix is unchanged. [Hint: Expand by the row (or column) being added to.]

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