Homework Answers
Question 3 (1 point) Which of the following statements are true? If one row of a...
Question 3 (1 point) Which of the following statements are true? If one row of a matrix is a linear combination of two other rows, then the determinant is 0. If the determinant of an nxn matrix is not zero, then the columns span the entire Rn The determinant is linear in each column. The row operation R2-R1-R2 (replacing row 2 by row 1 minus row 2) does not change the determinant. For all nxn matrices A and B, we...
2 invertible? C For which values of c is the matrix 8 O c 4 c...
2 invertible? C For which values of c is the matrix 8 O c 4 c =-4 Both of the above, i.e., c +4 Neither of the above, i.e., c +4. Suppose that the following row operations: interchange rows 1 and 3 multiply row 3 by 1/2 add -3 times row 1 to row 2 2 1 7 in this order, transform a matrix A into B = | 0 4-5 L0 0 3 What is the determinant of A?...
Problem 1 Consider the matrix Problem 1 Consider the matriz a 2 5 3 11 08...
Problem 1 Consider the matrix Problem 1 Consider the matriz a 2 5 3 11 08 a Find the cofactors C11,C2,C3 of A. b Find the determinant of 1, det(A) [ 2 4 61 Problem 2 Consider the matriz A=008 | 2 5 3 a Use the ero's to put A in upper triangular form 5 Pinul the determinant of A. (A) by keeping track of the row operations in part a and the properties of determinant Problem 3 Consider...
2. Determinant function onM 2 (a) Take A E M2. Consider the mapping volA: R2 x R2 - R, which is g...
ANSWER c d e ONLY! No need to answer a and b
Thanks
2. Determinant function onM 2 (a) Take A E M2. Consider the mapping volA: R2 x R2 - R, which is given by volA(v1, v2) olvA, 2A), for every v1, 02 E R2. Explain why volA is also a volume form (b) Explain why (use section (c) from question 1 above) there is a scalar α(A) E R such that VOLA-α(A) . voi We denote the scalar...
Let A and B be nxn matrices. Mark each statement true or false. Justify each answer....
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...
Answer these explanations: ADDING A MULTIPLE OF THE ith ROW TO THE jth row. 5.4 Example 6: Create a 5 by 5 matrix, E by...
Answer these explanations:
ADDING A MULTIPLE OF THE ith ROW TO THE jth row. 5.4 Example 6: Create a 5 by 5 matrix, E by typing: Type: ΕΞ11 2-134:10-1-2-1; 8 3 2 11:10-2-3-2:1112-1]. Find det(E) by typing: Type DE det(E) Note: Adding a multiple of ith row of E to the jth row in MATLAB can be done as follow. Values of i, j and k must be defined (entered) first. In the following line we choose i = 3,J...
Prove the following lemma. Let B be an n ✕ n matrix and let E be...
Prove the following lemma.
Let B be an n ✕ n
matrix and let E be an n ✕ n
elementary matrix. Then det(EB) = det(E)
det(B)
1. Write the proof and submit as a free response. (Submit a file
with a maximum size of 1 MB.)
2. Which of the following could begin a direct proof of the
statement?
If E interchanges two rows, then det(E) = 1 by
Theorem 4.4. Also, EB is the same as B but...
(3 points) Let A be a 4 x 4 matrix with det(A) = 8. 1. If...
(3 points) Let A be a 4 x 4 matrix with det(A) = 8. 1. If the matrix B is obtained from mes the second row to the first, then det(B) = 2. If the matrix C is obtained from A by swapping the first and second rows , then det(C) = 3. If the matrix D is obtained from A by multiplying the first row by 5, then det(D) =
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cof...
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cofactor expansion across the ith row of A, that is, det A H-1)adtAj Hint: Use induction on i, For the induction step from i to i+1, flip rows i and i+1 (How does this change the determinant?) and use the induction assumption.
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can...
Vetermine whether each statement is true or false. If a statement is true, give a reason...
Vetermine whether each statement is true or false. If a statement is true, give a reason or ote an appropriate statement from the text. If a statement is false provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. (a) The determinant of the sum of two matrices equals the sum of the determinants of the matrices. o, consider the following matrica ( 8 ) and (3) O...
Question 3 (1 point) Which of the following statements are true? If one row of a matrix is a linear combination of two other rows, then the determinant is 0. If the determinant of an nxn matrix is not zero, then the columns span the entire Rn The determinant is linear in each column. The row operation R2-R1-R2 (replacing row 2 by row 1 minus row 2) does not change the determinant. For all nxn matrices A and B, we...
2 invertible? C For which values of c is the matrix 8 O c 4 c =-4 Both of the above, i.e., c +4 Neither of the above, i.e., c +4. Suppose that the following row operations: interchange rows 1 and 3 multiply row 3 by 1/2 add -3 times row 1 to row 2 2 1 7 in this order, transform a matrix A into B = | 0 4-5 L0 0 3 What is the determinant of A?...
Problem 1 Consider the matrix Problem 1 Consider the matriz a 2 5 3 11 08 a Find the cofactors C11,C2,C3 of A. b Find the determinant of 1, det(A) [ 2 4 61 Problem 2 Consider the matriz A=008 | 2 5 3 a Use the ero's to put A in upper triangular form 5 Pinul the determinant of A. (A) by keeping track of the row operations in part a and the properties of determinant Problem 3 Consider...
ANSWER c d e ONLY! No need to answer a and b
Thanks
2. Determinant function onM 2 (a) Take A E M2. Consider the mapping volA: R2 x R2 - R, which is given by volA(v1, v2) olvA, 2A), for every v1, 02 E R2. Explain why volA is also a volume form (b) Explain why (use section (c) from question 1 above) there is a scalar α(A) E R such that VOLA-α(A) . voi We denote the scalar...
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...
Answer these explanations:
ADDING A MULTIPLE OF THE ith ROW TO THE jth row. 5.4 Example 6: Create a 5 by 5 matrix, E by typing: Type: ΕΞ11 2-134:10-1-2-1; 8 3 2 11:10-2-3-2:1112-1]. Find det(E) by typing: Type DE det(E) Note: Adding a multiple of ith row of E to the jth row in MATLAB can be done as follow. Values of i, j and k must be defined (entered) first. In the following line we choose i = 3,J...
Prove the following lemma.
Let B be an n ✕ n
matrix and let E be an n ✕ n
elementary matrix. Then det(EB) = det(E)
det(B)
1. Write the proof and submit as a free response. (Submit a file
with a maximum size of 1 MB.)
2. Which of the following could begin a direct proof of the
statement?
If E interchanges two rows, then det(E) = 1 by
Theorem 4.4. Also, EB is the same as B but...
(3 points) Let A be a 4 x 4 matrix with det(A) = 8. 1. If the matrix B is obtained from mes the second row to the first, then det(B) = 2. If the matrix C is obtained from A by swapping the first and second rows , then det(C) = 3. If the matrix D is obtained from A by multiplying the first row by 5, then det(D) =
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cofactor expansion across the ith row of A, that is, det A H-1)adtAj Hint: Use induction on i, For the induction step from i to i+1, flip rows i and i+1 (How does this change the determinant?) and use the induction assumption.
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can...
Vetermine whether each statement is true or false. If a statement is true, give a reason or ote an appropriate statement from the text. If a statement is false provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. (a) The determinant of the sum of two matrices equals the sum of the determinants of the matrices. o, consider the following matrica ( 8 ) and (3) O...
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