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  In linear algebra, a determinant is a scalar value associated with a square matrix. It provides important information about the properties of the matrix and plays a significant role in various mathematical and engineering applications. To calculate the determinant of a square matrix, there are different methods depending on the matrix's size. Here's a general overview: 2x2 Matrix : For a 2x2 matrix [a b; c d], the determinant is calculated as ad - bc. 3x3 Matrix : For a 3x3 matrix, you can use the expansion by minors or the Rule of Sarrus. It involves calculating the sum of products of certain elements in the matrix. Larger Matrices : For matrices larger than 3x3, you can use methods like cofactor expansion or row reduction to upper triangular form (Gaussian elimination) to calculate the determinant. Determinants have several key properties and applications, including: Determinants are used to determine whether a square matrix is invertible. If the determinant is nonzero, the...

  Matrices are rectangular arrays of numbers or other mathematical objects. They are widely used in various fields, including mathematics, physics, computer science, and engineering, to represent and solve systems of linear equations, transformations, and data structures. A matrix is typically denoted as an array enclosed in brackets, like this: A = [a11 a12 a13] [a21 a22 a23] Here, 'A' is the matrix name, 'a11', 'a12', etc., are the elements of the matrix, and the subscripts represent the row and column positions of each element. Matrices can be added, subtracted, multiplied, and manipulated in various ways to perform operations like matrix multiplication, matrix inversion, and solving linear equations. They are fundamental in linear algebra and have applications in computer graphics, statistics, and many other fields.

  Coordinates in three-dimensional space, often referred to as 3D coordinates, are used to locate a point within that space. Unlike two-dimensional coordinates, which use (x, y) values, 3D coordinates require three values (x, y, and z) to specify a point's position. Here's a brief explanation of 3D coordinates: X-axis : The x-coordinate represents the horizontal position of a point. It can be positive (to the right of the origin) or negative (to the left of the origin). Y-axis : The y-coordinate represents the vertical position of a point. It can be positive (above the origin) or negative (below the origin). Z-axis : The z-coordinate represents the depth or distance from the observer. It can be positive (away from the observer) or negative (toward the observer). The origin, where all three axes intersect, is typically denoted as (0, 0, 0). By specifying the values of x, y, and z, you can pinpoint any point in 3D space. For example, the point (2, -3, 1) is located 2 units to the...

  Three-dimensional geometry deals with objects and spaces in three dimensions. Some key concepts include: Coordinates : In a three-dimensional space, you use three coordinates (x, y, and z) to pinpoint a location. Vectors : Vectors are commonly used to represent direction and magnitude in 3D space. They can be added, subtracted, and multiplied by scalars. Distance and Length : You can calculate the distance between two points in 3D space using the 3D distance formula, which is an extension of the Pythagorean theorem. Lines and Planes : Understanding equations for lines and planes in 3D space is essential. Lines are represented by parametric equations, while planes have normal vectors and point representations. Dot and Cross Products : These operations are used extensively in 3D geometry. The dot product gives the projection of one vector onto another, while the cross product yields a vector perpendicular to both input vectors. Vectors and Parametric Equations of Curves : Curves li...

  A polynomial is a type of algebraic expression that consists of one or more terms, where each term is the product of a constant coefficient and one or more variables raised to non-negative integer exponents. The general form of a polynomial is: � ( � ) = � � � � + � � − 1 � � − 1 + … + � 2 � 2 + � 1 � + � 0 P ( x ) = a n ​ x n + a n − 1 ​ x n − 1 + … + a 2 ​ x 2 + a 1 ​ x + a 0 ​ Here's what each component represents: � ( � ) P ( x ) : The polynomial function. � � , � � − 1 , … , � 2 , � 1 , � 0 a n ​ , a n − 1 ​ , … , a 2 ​ , a 1 ​ , a 0 ​ : Coefficients. These are constants that multiply the corresponding terms. � x : The variable. � n : The highest non-negative integer exponent in the polynomial, known as the degree of the polynomial. For example, 2 � 3 − 3 � 2 + 5 � − 1 2 x 3 − 3 x 2 + 5 x − 1 is a polynomial of degree 3 because the highest exponent is 3, and it consists of four terms with coefficients 2, -3, 5, and -1. Polynomials are essential in algebra and mathematics, ...