19 For example A A 11A 12A 13A 21 A22 A23 A31 A det A A 11 A22 A23 A

19 for example a a 11a 12a 13a 21 a22 a23 a31 a det a

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being the indices of the row and column of that entry. [19] For example: A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 (52) det ( A ) = A 11 A 22 A 23 A 32 A 33 - A 12 A 21 A 23 A 31 A 33 + A 13 A 21 A 22 A 31 A 32 where the determinant is evaluated along the first row. It should be remarked that the [19] In fact, this rule for the determinant of an n × n matrix applies even to the 2 × 2 matrix if the cofactor of an entry in this case is taken as a single entry with the designated sign. However, we separated the 2 × 2 matrix case in the definition to be more clear and to avoid possible confusion.
1.3.3 Matrix Algebra 39 determinant of a matrix and the determinant of its transpose are equal, that is: det ( A ) = det ( A T ) (53) where A is a square matrix and T stands for the transposition operation. Another remark is that the determinant of a diagonal matrix is the product of its main diagonal elements. F. Inverse of Matrix The inverse of a square matrix A is a square matrix A - 1 where: AA - 1 = A - 1 A = I (54) with I being the identity matrix (see § Special Matrices) of the same dimensions as A . The inverse of a square matrix is formed by transposing the matrix of cofactors of the original matrix with dividing each element of the transposed matrix of cofactors by the determinant of the original matrix. [20] From this definition, it is obvious that a matrix possesses an inverse only if its determinant is not zero, i.e. it must be non-singular . It should be remarked that this definition includes the 2 × 2 matrices where the cofactor of an entry is a single entry with the designated sign, that is: A = A 11 A 12 A 21 A 22 A - 1 = 1 A 11 A 22 - A 12 A 21 A 22 - A 12 - A 21 A 11 (55) Another remark is that the inverse of an invertible diagonal matrix is a diagonal matrix obtained by taking the reciprocal of the corresponding diagonal elements of the original matrix. [20] The matrix of cofactors (or cofactor matrix) is made of the cofactors of its elements taking the same positions as the positions of these elements. The transposed matrix of cofactors may be called the adjugate or adjoint matrix although the terminology may differ between the authors.
1.4 Exercises 40 1.4 Exercises 1.1 Name three mathematicians accredited for the development of tensor calculus. For each one of these mathematicians, give a mathematical technical term that bears his name. 1.2 What are the main scientific disciplines that employ the language and techniques of tensor calculus? 1.3 Mention one cause for the widespread use of tensor calculus in science. 1.4 Describe some of the distinctive features of tensor calculus which contributed to its success and extensive use in mathematics, science and engineering. 1.5 Give preliminary definitions of the following terms: scalar, vector, tensor, rank of tensor, and dyad. 1.6 What is the meaning of the following mathematical symbols?

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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