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Mid2PracticeSol

# Mid2PracticeSol - Math 20F A00 Linear Algebra Spring 2011...

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Math 20F A00, Linear Algebra, Spring 2011 Practice Problems for Midterm Exam 2 1. Find all the co-factors, the adjugate matrix, and the determinant of the matrix 1 1 2 - 1 0 2 - 4 1 - 1 0 2 0 - 1 0 2 0 . 2. True or false: (a) Equivalent matrices have the same determinant; (F) (b) The determinant of any elementary matrix is 1; (F) (c) det( A + B ) = det A + det B , det( - A ) = - det A, and det( AB ) = (det A )(det B ); (F, F, T) (d) det A T = det A and det A - 1 = (det A ) - 1 ; (T, T) (e) A square matrix is invertible if and only if its determinant is nonzero. (T) 3. Calculate the determinant of each of the following matrices: 5 0 - 1 1 - 3 - 2 0 5 3 ; 1 1 2 - 1 0 0 - 4 0 - 1 3 12 0 1 2 - 4 1 ; 1 3 3 - 4 0 1 2 5 2 5 4 - 3 - 3 - 7 - 5 2 . 4. Show that det 1 1 1 a b c a 2 b 2 c 2 = ( a - b )( b - c )( c - a ). (Hint: transpose, row reduction, and expan- sion.) 5. Let A,B,C be three 2 × 2 matrices. Show that det bracketleftbigg A 0 B C bracketrightbigg = (det A )(det C ). 6. What is Cramer’s rule? Let A be a 3 × 3 matrix and assume det A = 2, the co-factors C 11 = 2, C 12 = - 4, C 13 = 6. What is the solution to

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