Label the following statements as true or false. (a) The function det: Mn×n(F) →F is a linear transformation. (b) The determinant of a square matrix can be evaluated by cofactor expansion along any row. (c) If two rows of a square matrix A are identical, then det(A) = 0. (d) If B is a matrix obtained from a square matrix A by interchanging any two rows, then det(B) = −det(A). (e) If B is a matrix obtained from a square matrix A by multiplying a row of A by a scalar, then det(B) = det(A). (f) If B is a matrix obtained from a square matrix A by adding k times row i to row j, then det(B) = k det(A). (g) If A ∈ Mn×n(F) has rank n, then det(A) = 0. (h) The determinant of an upper triangular matrix equals the product of its diagonal entries.
Label the following statements as true or false.
(a) The function det: Mn×n(F) →F is a linear transformation.
(b) The determinant of a square matrix can be evaluated by cofactor expansion along any row.
(c) If two rows of a square matrix A are identical, then det(A) = 0.
(d) If B is a matrix obtained from a square matrix A by interchanging any two rows, then det(B) = −det(A).
(e) If B is a matrix obtained from a square matrix A by multiplying a row of A by a scalar, then det(B) = det(A).
(f) If B is a matrix obtained from a square matrix A by adding k times row i to row j, then det(B) = k det(A).
(g) If A ∈ Mn×n(F) has rank n, then det(A) = 0.
(h) The determinant of an upper triangular matrix equals the product of its diagonal entries.
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