True or False? In Exercises 7578, determine whether each statement is true or false. If

Chapter 3, Problem 75

(choose chapter or problem)

In Exercises 75–78, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.

(a) The cofactor \(C_{22}\) of a matrix is always a positive number.

(b) If a square matrix B is obtained from A by interchanging two rows, then det(B) = det(A).

(c) If one column of a square matrix is a multiple of another column, then the determinant is 0 .

(d) If A is a square matrix of order n, then \(\operatorname{det}(A)=-\operatorname{det}\left(A^{T}\right)\)

Text Transcription:

C_22

det(A) = −det(A^T).