Math2JMidterm - then dividing the third column by 2. 4. Let...

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Math 2J, Spring 2011 Practice problems for the midterm 1. Determine whether the statement is true or false. (a) Overdetermined systems of equations always have no solution. (b) Row equivalent matrices have equal determinants (c) If zero is not an eigenvalue of the matrix A , then A is invertible. 2. (a) Give an example of an underdetermined system. (b) Give an example of an overdetermined system. (c) Give an example of augmented matrix for a system of 3 equations in 4 unknowns with infinitely many solutions. (d) Is there any system of 3 equations in 4 unknowns with a unique solution? If so, write down an example. 3. (a) If det( A ) = 4 and det( B ) = 5, what is det( AB )? What is det( A - 1 )? (b) If B is a 5 × 5 matrix with determinant 3, find the determinant of the matrix - 2 B . (c) If C is a 6 × 6 matrix with determinant 5, find the determinant of the matrix obtained by C by first switching the second and fourth row, and
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Unformatted text preview: then dividing the third column by 2. 4. Let A be a 5 5 matrix such that A =-A t . Compute det( A ). 5. Solve the following system of equations. ( Use any method you like.) x 1 + 2 x 2-4 x 3 = 0 , 3 x 1 + x 2 + 4 x 3 = 21 , 2 x 1 + 4 x 2-6 x 3 = 2 . 6. Compute the inverse of the matrix A = 1 1 1 2-3 1 1 3 . ( Use any method you like.) 7. Calculate the following determinant. det 1 1 1 1 1 2 1 1 1 1 4 1 1 1 1 3 8. Find the eigenvalues and eigenvectors of the matrix A = 1 0 1 1 . Is A diag-onalizable? 2 9. (a) Calculate the eigenvalues of the matrix A = 1 0 1 0 2 0 1 0 1 . (b) Find all eigenvectors of A . (c) Find an invertible matrix S and a diagonal matrix D such that S-1 AS = D. (d) Compute e A ....
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This note was uploaded on 03/11/2012 for the course MATH 2J 44360 taught by Professor Donaldson,neil during the Spring '10 term at UC Irvine.

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Math2JMidterm - then dividing the third column by 2. 4. Let...

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