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Chapter 2: Matrix Algebra

Expert-verified
Linear Algebra and its Applications
Pages: 93 - 164
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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311 Questions for Chapter 2: Matrix Algebra

  1. Unless otherwise specified, assume that all matrices in these exercises are \(n \times n\). Determine which of the matrices in Exercises 1-10 are invertible. Use a few calculations as possible. Justify your answer.

    Found on Page 93

  2. In Exercise 10 mark each statement True or False. Justify each answer.

    Found on Page 93

  3. Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}&{ - {\bf{3}}}\\{ - {\bf{4}}}&{\bf{6}}\end{aligned}} \right)\) and \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{8}}&{\bf{4}}\\{\bf{5}}&{\bf{5}}\end{aligned}} \right)\) and \(C = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}&{ - {\bf{2}}}\\{\bf{3}}&{\bf{1}}\end{aligned}} \right)\). Verfiy that \(AB = AC\) and yet \(B \ne C\).

    Found on Page 93

  4. Suppose A is invertible. Explain why \({A^T}A\) is also invertible. Then show that \({A^{ - {\bf{1}}}} = {\left( {{A^T}A} \right)^{ - {\bf{1}}}}{A^T}\).

    Found on Page 93

  5. Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).

    Found on Page 93

  6. In exercise 11 and 12, the matrices are all \(n \times n\). Each part of the exercise is an implication of the form “If “statement 1” then “statement 2”.”Mark the implication as True if the truth of “statement 2”always follows whenever “statement 1” happens to be true. An implication is False if there is an instance in which “statement 2” is false but “statement 1” is true. Justify each answer.

    Found on Page 93

  7. Let Abe an invertible \(n \times n\) matrix, and let B be an \(n \times p\) matrix. Show that the equation \(AX = B\) has a unique solution \({A^{ - 1}}B\).

    Found on Page 93

  8. Let \({x_1},...,{x_n}\) be fixed numbers. The matrix below called a Vandermonde matrix, occurs in applications such as signal processing, error-correcting codes, and polynomial interpolation.

    Found on Page 93

  9. In exercise 11 and 12, the matrices are all \(n \times n\). Each part of the exercise is an implication of the form “If “statement 1” then “statement 2”.”Mark the implication as True if the truth of “statement 2”always follows whenever “statement 1” happens to be true. An implication is False if there is an instance in which “statement 2” is false but “statement 2” is false but “statement 1” is true. Justify each answer.

    Found on Page 93

  10. Let Abe an invertible \(n \times n\) matrix, and let \(B\) be an \(n \times p\) matrix. Explain why \({A^{ - 1}}B\) can be computed by row reduction: If\(\left( {\begin{aligned}{*{20}{c}}A&B\end{aligned}} \right) \sim ... \sim \left( {\begin{aligned}{*{20}{c}}I&X\end{aligned}} \right)\), then \(X = {A^{ - 1}}B\).

    Found on Page 93

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