During the process of solving linear systems of equations, a new mathematical notation was introduced to simplify the writing involved. This notation introduced the concept of matrices, a term first coined by the English mathematician James Sylvester (1814 – 1897). Arthur Cayley (1821 – 1895) further developed the theory of matrices and applied them to linear transformations. In the modern era, matrices play a crucial role in high-speed computers and various other disciplines.
The concept of determinants, on the other hand, has historical roots in both Chinese and Japanese mathematics. However, the credit for the invention of determinants is attributed to the Japanese mathematician Seki Kowa (1642 – 1708) and the German mathematician Gottfried Wilhelm Leibniz (1646 – 1716). G. Cramer (1704 – 1752) successfully employed determinants in solving systems of linear equations.
A matrix is essentially a rectangular array of numbers enclosed by brackets, with horizontal lines representing rows and vertical lines representing columns. The individual numbers within the matrix are referred to as its entries or elements. Matrices can vary in size and order, with capital letters such as A, B, C, X, Y representing matrices and lowercase letters such as a, b, c denoting their entries.
A matrix’s order is specified as m % n, where m represents the number of rows and n represents the number of columns. Double subscripts, such as aij, are used to identify elements within matrices. For example, the element 7 is located at position a23 in the matrix. Matrices can contain elements from real numbers or complex numbers.
Additionally, matrices can exhibit various characteristics. A row matrix, or row vector, consists of a single row and multiple columns, while a column matrix, or column vector, has a single column and multiple rows. Rectangular matrices have unequal numbers of rows and columns, while square matrices have an equal number of rows and columns.
The principal diagonal of a square matrix consists of elements from the top left to the bottom right, and the secondary diagonal contains elements from the top right to the bottom left. A diagonal matrix has non-zero elements only on its principal diagonal, while a scalar matrix has non-zero elements only on its principal diagonal, which are all equal.
The unit matrix, or identity matrix, is a square matrix with ones on its principal diagonal and zeros elsewhere. A null matrix, or zero matrix, consists entirely of zeros.
Matrices can be added, subtracted, and multiplied, but the order of multiplication is important, as AB may not be equal to BA. Scalar multiplication is also defined, allowing matrices to be scaled by a scalar value.
Overall, matrices and determinants are fundamental concepts in mathematics, widely used in various applications, including computer science, physics, engineering, and many other fields.
Introductions of Chapter 3 Matrices and Determinants Notes
Exercise 3.1
Exercise 3.2
Exercise 3.3
Exercise 3.4
Exercise 3.5
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