Matrix is a rectangular arrangement of m × n numbers in the form of m horizontal lines and n vertical lines. These numbers can be real or complex. We call the horizontal lines rows and the vertical lines columns. As far as any entrance exam is concerned, matrices are an important topic. 2-3 questions can be expected from this topic for any entrance exam. So it is recommended that students should learn this topic thoroughly.
The rectangular array is enclosed by bracket [ ] or ( ). A matrix is denoted by A = [aij] mxn. Here a11, a12, ….. etc., are the elements of the matrix A, where aij belongs to the ith row and jth column and is called the (i, j)th element of the matrix A = [aij]. The three algebraic operations are involved in matrix operations. They are addition, subtraction, and multiplication. We can find the transpose of a matrix by changing the rows and columns of the matrix. We can denote the transpose of a matrix by AT. If A = [aij], then AT = [aji].
Different types of matrices are given below.
- Zero matrix
- Row matrix
- Column matrix
- Square matrix
- Singleton matrix
- Diagonal matrix
- Symmetric Matrix
- Skew-Symmetric Matrix
- Skew Hermitian Matrix
- Orthogonal matrix
- Idempotent matrix
- Nilpotent matrix
A matrix with all elements zero is called zero or null matrix. A row matrix contains only one row and a column matrix has only one column. A matrix with only one element is termed a singleton matrix. A matrix with an equal number of rows and columns is called a square matrix. A diagonal matrix is a square matrix in which all the elements except diagonal elements are zeros.
The three basic operations on the matrix are addition, subtraction, and multiplication. For addition and subtraction, the order of the matrix should be identical. Let A and B be two matrices. To multiply A and B, the number of columns in the A should be equal to the number of rows in B. Multiplication is not commutative in the case of matrices. Addition is commutative in the case of a matrix. This means A+B = B+A.
Permutations deal with arranging those items in a definite order. The selection of items from a group of items is termed as combinations. In permutation, order matters. In short, we can say that combination is about selection and permutations is about the arrangement of objects without actually listing them. We can select items in any order, in combination.
Two basic principles of counting are the fundamental principle of counting and addition principle. As per the fundamental principle, if the event P occurs in n different ways and another event Q occurs in m different ways, then the total number of occurrences of two events is = m x n. According to the addition principle, if an event P occurs in m different ways and event Q occurs in n different ways and both the events cannot occur together, then the occurrence of events P or Q is given by m + n. Visit BYJU’S for more information regarding matrix, permutation, and combination.