Exam1_review - Review Sheet for Exam 1 Math 291 Definitions...

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Unformatted text preview: Review Sheet for Exam 1 - Math 291 Definitions: Elementary row operation, elementary matrix, echelon form, reduced echelon form, augmented matrix, free variable, leading variable,inverse of A, transpose of A, determinant of a square matrix, singular and non-singular matrices, linear combination, general solution, particular solution, consistent/inconsistent system of equations Things you should know how to do: How to solve a system of linear algebraic equations by reducing the augmented matrix to echelon form How to identify the leading variables and the free ones How to write out the general solution in vector form How to do all of the above with elementary matrices, in principle. How to recognize inconsistent systems, systems with nonunique soltuions, etc. How to compute a determinant using row operations Properties of the determinant that follow from the definition. Some sample (non-computational) questions: 1. For Ax = b, where A is 4 6: (T/F) The system is always consistent (T/F) The homogeneous system is always consistent. (T/F) The homogeneous system has nontrivial solutions. (You should be able to give reasons or counterexamples to support each of your answers above.) 2. Any matrix A can be written as the product A = BR, where B is invertible and R is upper triangular. (Hint: Think about reducing A to echelon form (not reduced echelon form) using elementary matrices .) 3. If rowi (A) = rowj (A) + rowk (A), i = j = k, then det(A) = 0. 4. Suppose A is 5 3, and that the solution to the homogeneous equation has one free variable. How many leading variables are there? Do you expect that, in general, the inhomogeneous system corresponding to this matrix will have solutions? Will it have solutions for some right hand sides b? 5. Show that any invertible matrix can be written as the product of elementary matrices. 6. Is it possible that, given a non-square matrix A, the equation Ax = b has a unique solution for any b? Explain your answer. 1 ...
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