Question

Let A and B be nxn matrices. Mark each statement true or false. Justify each answer. Complete parts (a) through (d) below. a. The determinant of A is the product of the diagonal entries in A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The statement is false because the determinant of the 2x2 matrix A = is not equal to the product of the entries on the main diagonal of A. (Type an integer or simplified fraction for each matrix element.) O B. The statement is true because the determinant of any triangular matrix A is the product of the entries on the main diagonal of A. O c. The statement is true because the determinant of any square matrix A is the product of the entries on the main diagonal of A. b. An elementary row operation on A does not change the determinant. Choose the correct answer below. O A. The statement is false because scaling a row also scales the determinant by the same scalar factor. OB. The statement is true because the effects of interchanging two rows cancel out during a determinant calculation. O c. The statement is false because a row replacement operation on A changes the sign of the determinant. OD. The statement is true because the determinant depends on the reduced echelon form of a matrix, which is the same regardless of the row operations used to generate it. C. (det A)(det B) = det AB. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The statement is false because the correct property is det A + det B = det(A + B). OB. The statement is true because it is the Multiplicative Property of determinants. O C. The statement is true because the product of the determinants is equal to the determinant of the product for the two 2x2 matrices (Type an integer or simplified fraction for each matrix element. Use a comma to separate answers as needed.) OD. The statement is false because the product of the determinants is not equal to the determinant of the product for the two 2x2 matrices (Type an integer or simplified fraction for each matrix element. Use a comma to separate answers as needed.) d. If 2 + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The statement is false because the factors of the characteristic polynomial are useful for determining the eigenvectors of a matrix, not its eigenvalues. O B. The statement is true because if + 5 is a factor of the characteristic polynomial of A, then 1 +5 must be an entry on the main diagonal of A-21. OC. The statement is false because in order for 5 to be an eigenvalue of A, the characteristic polynomial would need to have a factor of (Type an expression using 1 as the variable.)

Answer 1

у до on A Of A, the characteritic B. The Statement is true because the determinant of ang traingular matrix A 19 the broduct 07 the entries of the main diagonal of A Ы (А) The statement 18 fat se because slating scates the determinant by the same scapar factor ③ The statement is takse because a row reb Pacement operation changes the sign ol the determinant @ The statement is tapse becuase the product of determinants is not equal to the determinant! two of broduct of 2x2 matrices A 3 y ABE 3 det (AB) = -2 det (A) = -2 . det (B) = .. det lA).det (B) = det (AB) @ @ The statement is topse because tor 5 to be eigen value bokynomiat would need to have [83][6:1-15; in order a tector of (d-5).