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Sample+problem+set+for+the+Midterm

# Sample+problem+set+for+the+Midterm - then dividing the...

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Math 2J, Fall 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 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. 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 . 5. Compute the inverse of the matrix A = 1 1 1 2-3 1 1 3 . ( Use any method you like.) 6. Calculate the following determinant. det 1 1 1 1 1 2 1 1 1 1 4 1 1 1 1 3 7. Find the eigenvalues and eigenvectors of the matrix A = ± 1 0 1 1 ² . Is A diag-onalizable? 2 8. (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....
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Sample+problem+set+for+the+Midterm - then dividing the...

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