The determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle To compute the inverse of a matrix, the determinant is required. For each square matrix A, there is a unit scalar value known as the determinant of … Read more
An invertible matrix A is called a row equivalent to an identity matrix, and we can this matrix by understanding the row reduction of A to I. A n x n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations … Read more
An elementary matrix is expressed by performing a single elementary row operation on an identity matrix.The operation on a row is denoted by the notation Ri and column operation is denoted using the notation Cj while k represents the scalar quantity. There are three types of elementary row operations: 1. Interchange two rows or columns … Read more
Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrices (restrict to square matrices of order 2) Matrix multiplication is not commutative: AB ≠ BA. Example: Consider the following example, calculate AB and BA Because A has a dimension of 2 x 2 and B has a dimension of 2 … Read more
Commutative Law: A + B = B + A The definition of subtracting two real numbers a and b is a – b = a + (-1)b or a + the opposite of b. We can define subtraction of matrices similarly, therefore,when A and B are two matrices of the same dimensions, then A – B … Read more
Operation on matrices: Addition and multiplication and multiplication with a scalar For adding and subtracting matrices, they must have the same order, mxn. To add matrices of the same order, add their corresponding elements and to subtract matrices of the same order, subtract their corresponding elements. The following mathematical notation can be used: Note that … Read more
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew-symmetric matrices A matrix is a set or group of numbers that are arranged in a rectangular array or a square enclosed in two brackets. A matrix is expressed by a bold capital letter and the elements are … Read more
The elementary properties of inverse trigonometric functions are valid within the principal value branches of the corresponding inverse function wherever they are defined. However, these properties are valid for a limited section of the domain of the inverse functions. Recall that, if y = sin$^{-1}$x and x= sin y then y = sin$^{-1}$x. This means … Read more
The graph of y = sin x can be visuallised in the figure below: Domain: all reals Range: [-1,1] Period: 2π Y-intercept: (0,0) When you restrict the domain of sin x to the interval –π/2 ≤ x ≤ π/2, the following properties should hold: 1.y = sin x is an increasing function. 2.y = sin … Read more
Recall from previous chapters that inverse of a function f can be written as f-1, where the expression sin-1 is read as “the inverse sine.” In this notation,-1 indicates the inverse and not the reciprocal of the sine function.Other names for inverse trigonometric functions are arcsine, arccosine, and arctangent functions. The inverses are not functions … Read more