Geometric Progression’s general term with “a” as the first term and “r” as the common ratio is defined by: aₙ = a.rⁿ⁻¹ Where ‘n’ = the number of terms. Theorem: To prove that G.P.’s nth term with “a” as the first term & “r” as the common ratio which is given by: aₙ = a.rⁿ⁻¹ … Read more

The binomial theorem or the binomial expansion is loaded with its extremely large applications and is immensely beneficial in the simplification of the lengthy calculations. The binomial theorem or the expansion for the nth polynomial degree is given by: If there’s a need for the computation of (1+x)¹² which doesn’t mean that you to multiply … Read more

Recall that a cofactor matrix C of a matrix A is the square matrix of the same order as A in which each element aij is replaced by its cofactor cij . The adjoint matrix of [A] is written as Adj[A] and it can be obtained by obtaining the transpose of the cofactor matrix of … Read more

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