A. determine whether the statement "if a is 3 times 3, with columns a_1, a_2, a_3, then det a equals the volume of the parallelepiped determined by a_1, a_2, a_3" is true or false. choose the correct answer below. a. the statement is true because if m is a 2 times 2 matrix, then the area of the parallelogram determined by the columns of b is |det b|. the given statement is the natural extension of this to 3 times 3 matrices. b. the statement is false because |det a| equals the volume of the parallelepiped determined by a_1, a_2, a_3. it is possible that |det a| notequalto det a. c. the statement is false because it is possible to have det a = 0. when this happens, det a cannot be the volume of a parallelepiped since volume must be positive. d. the statement is true because | det a| equals the volume of the parallelepiped determined by a_1, a_2, a_3. because det a greaterthanorequalto 0 for all 3 times 3 matrices a, it follows that | det a| = det a. b. determine whether the statement "det a^t = (- 1)det a" is true or false. choose the correct answer below. a. the statement is true because a^t = a for any n times n matrix a. b. the statement is true because a row interchange is needed to turn a matrix into its transpose. a row interchange changes the sign of the determinant. c. the statement is false because det a^t = (-1)^n det a for any n times n matrix a. d. the statement is false because det a^t = det a for any n times n matrix a. c. determine whether the statement "the multiplicity of a root r of the characteristic equation of a is called the algebraic multiplicity of r as an eigenvalue of a" is true or false. choose the correct answer below. a. the statement is false because the multiplicity of a root r of the characteristic equation of a is called the geometric multiplicity of r as an eigenvalue of a. b. the statement is true because it is the definition of the algebraic multiplicity of an eigenvalue of a. c. the statement is false because the multiplicity of a root r of the characteristic equation of a is the number of eigenvectors corresponding to that root. d. the statement is true because it is the definition of the multiplicity of a root of the characteristic equation of a. d. determine whether the statement "a row replacement operation on a does not change the eigenvalues" is true or false. choose the correct answer below. a. the statement is true because a row replacement operation on a does not change the determinant of a. if two matrices have the same determinant, then they have the same eigenvalues. b. the statement is true because row equivalence is the same as similarity. if two matrices are similar, then they have the same eigenvalues. c. the statement is alse because row operations on a matrix usually change its eigenvalues. d. the statement is false because row operations on a matrix change the algebraic multiplicities of its eigenvalues.