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PODASIP the following statements about products and inverses of triangular and unit triangular matrices. a. The inverse of a triangular matrix is triangular. b. The product of two triangular matrices is triangular. c. The inverse of a unit upper (lower) triangular matrix is upper (lower) triangular. d. The product of two-unit triangular matrices is unit triangular.

PODASIP the following statements about products and inverses of triangular and unit triangular matrices. a. The inverse of a triangular matrix is triangular. b. The product of two triangular matrices is triangular. c. The inverse of a unit upper (lower) triangular matrix is upper (lower) triangular. d. The product of two-unit triangular matrices is unit triangular.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Triangular and Unit Triangular Matrices: Properties and Behaviors** In this section, we will explore properties and behaviors related to triangular and unit triangular matrices, particularly focusing on their products and inverses. Consider the following statements: **a. The inverse of a triangular matrix is triangular.** If \( A \) is a triangular matrix, then \( A^{-1} \) (if it exists) will also be a triangular matrix. This property simplifies many computations and solutions in linear algebra, maintaining structural form and reducing complexity. **b. The product of two triangular matrices is triangular.** When multiplying two triangular matrices of compatible dimensions, the resulting matrix will also be triangular. For instance, if \( A \) and \( B \) are both upper triangular matrices, their product \( AB \) will still be an upper triangular matrix. **c. The inverse of a unit upper (lower) triangular matrix is upper (lower) triangular.** For unit triangular matrices (triangular matrices with all diagonal entries equal to 1): - The inverse of a unit upper triangular matrix is also an upper triangular matrix. - The inverse of a unit lower triangular matrix is also a lower triangular matrix. This maintains the unit triangular structure and can be particularly useful in computational methods such as solving systems of linear equations. **d. The product of two unit triangular matrices is unit triangular.** When multiplying two unit triangular matrices: - The resulting matrix remains unit triangular. This property ensures that the diagonal entries remain 1, preserving the unit triangular nature through multiplication operations. **Explanation:** - **Triangular Matrix**: A matrix is triangular if it is either upper triangular (all elements below the main diagonal are zero) or lower triangular (all elements above the main diagonal are zero). - **Unit Triangular Matrix**: A triangular matrix where all the entries on the main diagonal are 1. Understanding these properties aids in simplifying many matrix operations, contributing to more efficient computational algorithms and a deeper grasp of linear algebraic structures.
Transcribed Image Text:**Triangular and Unit Triangular Matrices: Properties and Behaviors** In this section, we will explore properties and behaviors related to triangular and unit triangular matrices, particularly focusing on their products and inverses. Consider the following statements: **a. The inverse of a triangular matrix is triangular.** If \( A \) is a triangular matrix, then \( A^{-1} \) (if it exists) will also be a triangular matrix. This property simplifies many computations and solutions in linear algebra, maintaining structural form and reducing complexity. **b. The product of two triangular matrices is triangular.** When multiplying two triangular matrices of compatible dimensions, the resulting matrix will also be triangular. For instance, if \( A \) and \( B \) are both upper triangular matrices, their product \( AB \) will still be an upper triangular matrix. **c. The inverse of a unit upper (lower) triangular matrix is upper (lower) triangular.** For unit triangular matrices (triangular matrices with all diagonal entries equal to 1): - The inverse of a unit upper triangular matrix is also an upper triangular matrix. - The inverse of a unit lower triangular matrix is also a lower triangular matrix. This maintains the unit triangular structure and can be particularly useful in computational methods such as solving systems of linear equations. **d. The product of two unit triangular matrices is unit triangular.** When multiplying two unit triangular matrices: - The resulting matrix remains unit triangular. This property ensures that the diagonal entries remain 1, preserving the unit triangular nature through multiplication operations. **Explanation:** - **Triangular Matrix**: A matrix is triangular if it is either upper triangular (all elements below the main diagonal are zero) or lower triangular (all elements above the main diagonal are zero). - **Unit Triangular Matrix**: A triangular matrix where all the entries on the main diagonal are 1. Understanding these properties aids in simplifying many matrix operations, contributing to more efficient computational algorithms and a deeper grasp of linear algebraic structures.
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