Introduction
Matrices and determinants are two related mathematical concepts used in linear algebra. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The numbers, symbols, or expressions in the matrix are known as its elements or entries.
A determinant is a number that is derived from the elements in a matrix and it is used to describe the properties of the matrix. Determinants are used to solve systems of linear equations, calculate the inverse of a matrix, and calculate the volume of a parallelepiped.
Some Special Types of Matrices
In class 10, we have learned only about 4 types of matrix but in class 11 we have to study about 6 different types of matrices. Let’s have a quick look at the type of matrices.
- Square Matrix
- Unit Matrix or Identity Matrix
- Zero Matrix or Null Matrix
- Triangle Matrix
- Symmetric Matrix
- Skew-symmetric Matrix
Transpose of a Matrix
The new matrix obtained from a given matrix A by interchanging its rows and columns is called the transpose of A. It is denoted by A’ or Aᵀ. You can just remember the format of finding the transpose of the given matrix and it will be easier for you to solve any problem of same kind.
Exercise – 5.1
Introduction
As mentioned before, Matrices and determinants are two related mathematical concepts used in linear algebra. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns whereas determinant is a number that is derived from the elements in a matrix and it is used to describe the properties of the matrix.
Matrices and determinants chapter helps students in understanding the concept of linear equations and their solutions. It will hwlp you to learn how you can make use of matrices in solving systems of linear equations. In this chapter, we will learn about different properties of matrices and determinants, which are essential in solving problems related to matrices and determinants.
Determinant
Determinant originally appeared in the study of linear equations. We shall, however, associate the notion of the determinant with matrices. For this purpose, we have to consider square matrices of order 2, 3 ,…,n, i.e.2×2, 3×3,… ,n×n matrices.
Properties of Determinants
We have seen how to evaluate a determinant. Evaluation becomes easy with the help of the following properties. We shall simply verify these properties.
- The value of the determinant is unaltered by interchanging its rows and columns.
- Interchanging any two adjacent rows (or columns) changes the sign of the determinant.
- If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
- If all the elements of any row (or column) are multiplied by a constant k, then the value of the determinant is multiplied by k.
- If each element of any row (or column) of a determinant is expressed as the sum of two terms, then the determinant can be expressed as a sum of two determinants.
- If two the elements of any row (or column) a multiple of any other row (or column) is added, the value of the determinant remains unaltered.
Exercise – 5.2
Introduction
As mentioned before, Matrices and determinants are used in linear algebra and it is defined a rectangular array of numbers, symbols, or expressions arranged in rows and columns whereas determinant is a number that is derived from the elements in a matrix and it is used to describe the properties of the matrix.
Matrices and determinants chapter helps students in understanding the concept of linear equations and their solutions. It will hwlp you to learn how you can make use of matrices in solving systems of linear equations. In this chapter, we will learn about different properties of matrices and determinants, which are essential in solving problems related to matrices and determinants.
Adjoint of Matrix
If you want to solve all problems of this exercise then you should be well known about adjoint of matrix. To understand it in simple language, the adjoint of a matrix is the transpose of its cofactor matrix and the cofactor matrix is obtained by taking the determinant of the minor matrix for each element of the original matrix. The adjoint matrix can be used to compute the inverse of a matrix, if it exists.
Inverse of Matrix
In linear algebra, the inverse of a matrix is a matrix that, when multiplied with the original matrix, produces the identity matrix. It is denoted by A⁻¹, where A is the original matrix. The inverse of a matrix is used to solve linear equations. For a matrix to have its inverse the given matrix should be a square matrix and the determinant of that matrix should not be equal to zero.
