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Mangal, hi. I have another one homework for linear algebra. Can you take a look ?

1. An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a dollar's worth of agriculture requires inputs of $0.30 from agriculture, $0.30 from manufacturing and $0.20 from energy. Production of a dollar's worth of manufacturing requires inputs of $0.20 from agriculture, $ 0.20 from manufacturing, and $0.30 from energy. Production of a dollar's worth of energy requires inputs of $0.30 from agriculture, $0.40 from manufacturing, and $0.30 from energy. Find the output for each sector that is needed to satisfy a final demand of $54 billion for agriculture, $29 billion for manufacturing, and $36 billion for energy. The output of the agricultural sector is _______ billion dollars. 2. Prove the theorem The (i,j)-entry of is the (j,i)-entry of AB, which is ______________________ 3. compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice. a. using this expansion, the determinant is -(0)(-6)+(4)(-4)-(5)(-8)=? b. using this expansion, the determinant is -(2)(-18)+(0)(-6)-(4)(15)=? c. using this expansion, the determinant is (2)(-18)-(0)(-6)+(4)(15)=? d. using this expansion, the determinant is (0)(-6)-(4)(-4)+(5)(-8)=? 4. compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice. a. using this expansion, the determinant is -(3)(-13)+(-3)(-6)-(4)(11)=? b. using this expansion, the determinant is (3)(-13)-(-3)(-6)+(4)(11)=? c. using this expansion, the determinant is (-3)(-6)-(1)(-7)+(4)(-3)=? d. using this expansion, the determinant is -(-3)(-6)+(1)(-7)-(4)(-3)=? 5. Let A be an n×n matrix. Mark each statement True or False. Justify each answer.
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a. The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row. b. the determinant of a triangular matrix is the sum of the entries on the main diagonal. a. choose the correct answer below A. false, because the determinant of A can only be calculated by cofactor expansion across a row. Cofactor expansion down a column has no relation to the determinant. B. true, because the plus or minus sign of the (i,j)-cofactor depends on the position of in matrix A. cofactor expansion down a column switches the order of i and j. thereby switching the sign of the cofactor expansion across a row. C. false, because the determinant of A can be computed by cofactor expansion across any row or down any column. since the determinant of A is well defined, both of these cofactor expansions will be equal. D. true, because cofactor expansion across a row adds each of the cofactors together. cofactor expansion down a column subtracts each cofactor from one another, this causes the two cofactor expansions to have opposite signs. 6. If The determinant is _____ 7. Use determinants to find out if the matrix is invertible The determinant of the matrix is ______ (simplify your answer) 8. find The determinant by row reduction to echelon form Row reduce the given matrix to echelon form 9. find The determinant by row reduction to echelon form Row reduce the given matrix to echelon form 10. compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix
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Subject: Linear Algebra, Math

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