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F10PracMT2

# F10PracMT2 - M341 Fall 10 Practice Midterm 2 First...

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M341, Fall 10. Practice Midterm 2. First Name ......................... Last Name (PRINT) ............................... - Calculators are authorized. Books, hand written notes or other documents are not authorized. - Show all your work. If you use a calculator, you are not waived of the intermediary computations that are still to be shown. - Explain every answer, either by a computation, or by using a theorem. Cor- rect results with no explanation will not be taken into account. - Freely use both sides of the sheets. - Put your name on any additional or unstapled sheet. Problem 1. 1) Compute the determinant of the matrix: A = 3 2 1 1 6 3 2 4 0 . Is A invertible and why? 2) What is the adjoint or adjugate matrix of A ? What is the inverse of A ? (Show all your work. A correct explanation of what is a cofactor and what is the adjoint matrix might redeem small computational mistakes.) 3) Compute the inverse of A by a row reduction method. 4) Solve the linear system (You have the choice of the method, but explain your choice): 3 x 1 + 2 x 2 x 3 = 6 x 1 + 6 x 2 + 3 x 3 = 30 2 x 1 4 x 2 = 8 Solution 1

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3 2 1 1 6 3 2 4 0 = 0 16 10 1 6 3 0 16 6 = 4 0 8 5 1 6 3 0 8 3 = 4( 1) × 8 1 5 1 3 = ( 1)[4 × ( 16)] = 64 . Since det A ̸ = 0, the matrix A is invertible. The matrix of minors ( A i,j instead of a i,j ) is: 12 6 16 4 2 16 12 10 16 No need of a calculator, all the matrices are 2x2 with small integers as entries. Of course the less confident you are in your computations, the more intermediaries you must show for partial credit. Anyway, results without a suﬃcient explanation on how you got them will certainly not provide you with the full grade even if corret. The matrix of cofactors is obtained by replacing the A i,j by the ( 1) i + j A i,j : 12 6 16 4 2 16 12 10 16 . Finally the Adjugate or Adjoint matrix is the transpose of the matrix of cofactors. A = 12 4 12 6 2 10 16 16 16 . In this problem, you are asked to give the adjoint matrix so the inverse must be computed by the formula A - 1 = 1 det A Adj( A ). The row reduction method would not be relevant since you know A .
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