# HSC Higher Mathematics 1st Paper Note 1st Chapter Matrices and Determinants

HSC Higher Mathematics 1st Paper Note 1st Chapter Matrices and Determinants. This article is about the determinants of mathematics. For determinants in epidemiology, see risk factor. In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. It can be viewed as the scaling factor of the transformation described by the matrix. In the case of a 2 × 2 matrix, the specific formula for the determinant is:

The Determinant of a Matrix. DEFINITION: Determinants play an important role in finding the inverse of a matrix and also in solving systems of linear equations. In the following, we assume we have a square matrix (m = n). The determinant of a matrix A will be denoted by det(A) or. In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or. | A

## HSC Higher Mathematics 1st Paper Note 1st Chapter Matrices and Determinants

Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations, although more efficient techniques are actually used, some of which are determinant-revealing and consist of computationally effective ways of computing the determinant itself. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables.

Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. In analytic geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down. When the entries of the matrix are taken from a field (like the real or complex numbers), it can be proven that any matrix has a unique inverse if and only if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.

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