Table of Contents
What are Determinants and Matrices?
In mathematics, a determinant is a function that takes a square matrix as input and outputs a scalar. The determinant of a matrix is calculated by multiplying the elements of the matrix along the main diagonal and then subtracting the product of the elements off the diagonal.
The determinant of a matrix can be used to solve systems of linear equations.
A matrix is a two-dimensional array of numbers. The numbers in a matrix can be used to represent the coefficients of a system of linear equations.
Types of Matrices
A matrix is a rectangular array of numbers, symbols, or other mathematical objects. The numbers in a matrix are called its entries, or its elements. Matrices can be square, that is, the number of rows and columns are the same, or they can be rectangular, that is, the number of rows can be different from the number of columns.
There are three types of matrices:
1. Diagonal matrices have all of the off-diagonal entries equal to zero.
2. Symmetric matrices are diagonal matrices in which the diagonal entries are all equal.
3. Hermitian matrices are symmetric matrices in which the diagonal entries are all complex numbers.
Operations on Matrices
There are a number of basic operations that can be performed on matrices, including addition, subtraction, multiplication, and division.
Addition and subtraction are fairly straightforward – the only thing to keep in mind is that matrix addition and subtraction are performed column-wise, not row-wise. For example, the following matrices are all addition and subtraction examples:
\[\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} + \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} = \begin{bmatrix} 8 & 10 \\ 12 & 14 \\ 16 & 18 \end{bmatrix}\]
\[\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} – \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} = \begin{bmatrix} -6 & -2 \\ -4 & 0 \\ -2 & 2 \end{bmatrix}\]
Multiplication is a bit more complicated, as it can be performed in two different ways: matrix multiplication or element-wise multiplication.
Matrix multiplication is the standard way to multiply two matrices, and is performed by multiplying the corresponding entries in the two matrices. For example, the
on Matrix Operations:
1) Addition:
The addition of two matrices is performed by adding the corresponding elements of the two matrices.
For example, if A = [1, 2, 3] and B = [4, 5, 6], then the addition of A and B is:
A + B = [5, 7, 9]
2) Matrix Multiplication:
The multiplication of two matrices is performed by multiplying the corresponding elements of the two matrices.
For example, if A = [1, 2, 3] and B = [4, 5, 6], then the multiplication of A and B is:
A*B = [16, 30, 36]
Determinant of a Matrix
A determinant is a mathematical function that is used to calculate the size of a certain term in a matrix. It is a scalar value, meaning that it has a single numerical value. The determinant of a matrix can be used to calculate certain characteristics of the matrix, such as its rank and inverse.
Solved Example:
A student has a weight of 54 kg and height of 1.6 m. What is the student’s BMI?
The student’s BMI is 22.5 kg/m2.
Properties of Determinant
The determinant of a square matrix is a real number that is associated with the matrix. The determinant is found by taking the product of the diagonal elements and then subtracting the product of the off-diagonal elements.
The determinant is also equal to the volume of the parallelepiped formed by the columns of the matrix.
Significance of Matrices and Determinants in Mathematics
Matrices and determinants are important in mathematics because they are used in a variety of mathematical applications. Matrix algebra is used in solving problems in physics and engineering, while determinants are used in solving problems in physics, engineering, and mathematics.
Significance of Matrices in Business Administration
Matrices are used in business administration to represent relationships between objects. For example, a company might use a matrix to show the relationships between products, customers, and sales. Matrices can also be used to represent the financial relationships between companies.
History of Determinants and Matrices
The history of determinants and matrices is a long and complex one. The origins of determinants can be traced back to 17th century France, while the origins of matrices can be traced back to 18th century India. Over the years, there have been many mathematicians who have made significant contributions to the development of determinants and matrices. Some of the most notable mathematicians include Pierre de Fermat, Leonhard Euler, Carl Friedrich Gauss, and Arthur Cayley.
