In linear algebra, the transpose of a matrix is an operation that flips the matrix over its diagonal, reflecting its rows and columns. The transpose of a matrix A is denoted by A^T. To compute the transpose of a matrix, you simply write the rows of the matrix as columns, and the columns as rows.…
Multiplication by a scalar: Multiplying a matrix by a scalar means multiplying every entry of the matrix by that scalar. For example, if A is a matrix and k is a scalar, then the product kA is obtained by multiplying every entry of A by k. Formally, if A = [a_ij] is an m x…
Matrices addition is a mathematical operation that involves adding two matrices of the same size element-wise. Given two matrices A and B, the sum C=A+B is obtained by adding the corresponding elements of A and B: C(i,j) = A(i,j) + B(i,j) where C(i,j), A(i,j), and B(i,j) are the elements located at the i-th row and…
Two matrices are said to be equal if they have the same dimensions and their corresponding entries are equal. In other words, if A = [aij] and B = [bij] are two matrices of the same size, then A and B are equal if and only if aij = bij for all i and j.…
A matrix is a rectangular array of numbers, which can be real numbers, complex numbers, or even other mathematical objects such as polynomials. In the case of matrices consisting of real numbers, each entry in the matrix is a real number. A matrix can be represented as a rectangular array of numbers enclosed by brackets.…
Binomial coefficients, also known as “choose” coefficients, are mathematical objects that arise in the study of combinatorics and probability theory. They are denoted by the symbol ${n\choose k}$, and represent the number of ways to choose k objects from a set of n distinct objects, without regard to their order. There are several important properties…
The Binomial Theorem is a formula that provides a way to expand expressions of the form: (a + b)^n where “n” is a positive integer and “a” and “b” are any real numbers. The formula is given by: (a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ……
Permutations and combinations are concepts in combinatorics, which is the branch of mathematics concerned with counting and arranging objects. A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time, denoted by P(n,r), is given by: P(n,r) = n!/(n-r)! where n! represents n…
Logarithms are mathematical functions that are used to solve equations involving exponents. They are defined as the inverse of exponential functions. In other words, if we have an exponential function, a^x, the logarithm of that function, written as loga(x), is the exponent to which we raise the base a to get x. The properties of…
Here are the formulas for the sums of squares and cubes of the first n natural numbers: Sum of squares: 1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6 Sum of cubes: 1³ + 2³ + 3³ + … + n³ = (n(n+1)/2)² To understand how these formulas are derived, let’s look at…