(Linear algebra)Pls answer question asapWrite work neatly and box final answersThanks **Matrix Determinant Problems**Suppose \( A \) and \( B \) are \( 4 \times 4 \) matrices with \( |A| = 4 \) and \( |B| = 5 \), answer the following. If the answer is unknown, write UNKNOWN.1. \( \det \left( A^T \right) = \) [ ]2. \( \det (AB) = \) [ ]3. \( \det \left( B^{-1} A \right) = \) [ ]4. \( \det (B - A) = \) [ ]5. \( \det(3A) = \) [ ]**Instructions**: Please calculate the determinants based on the given properties of matrices and fill in the blanks. If you are unsure about any calculation, mark it as "UNKNOWN".

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Answered: Suppose A and B are 4 x 4 matrices with…

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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**Matrix Determinant Problems**

Suppose \( A \) and \( B \) are \( 4 \times 4 \) matrices with \( |A| = 4 \) and \( |B| = 5 \), answer the following. If the answer is unknown, write UNKNOWN.

1. \( \det \left( A^T \right) = \) [ ]

2. \( \det (AB) = \) [ ]

3. \( \det \left( B^{-1} A \right) = \) [ ]

4. \( \det (B - A) = \) [ ]

5. \( \det(3A) = \) [ ]

**Instructions**: Please calculate the determinants based on the given properties of matrices and fill in the blanks. If you are unsure about any calculation, mark it as "UNKNOWN".

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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