Determinants
Chapter 4 of 12 Class NCERT Maths Solution discusses the topic of determinants. Students will get to learn about the definition and meaning of determinants, remarks based on the order of determinants, properties of determinants, finding the area of a triangle using determinants, minors and cofactors of determinants, adjoint of a matrix, the inverse of a matrix, applications of determinants, and matrices and miscellaneous examples. Below we have links provided to each exercise solution covered in this chapter. 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. Adjoint and inverse of a square matrix. Consistency, inconsistency, and the number of solutions of a system of linear equations by examples, solving a system of linear equations in two or three variables (having unique solution) using the inverse of a matrix.
Determinant of a Matrix
The determinant is the numerical value of the square matrix. So, to every square matrix A = [aij] of order n, we can associate a number (real or complex) called the determinant of the square matrix A. It is denoted by det A or |A|.
e.g. The determinant of a matrix A = [a11]1 × 1 can be given as: |a11| = a11.
Before finding the Determinant of a Matrix we must learn the Minor and Cofactor of a Matrix.
Minor
Minor of an element ay of a determinant is a determinant obtained by deleting the ith row and jth column in which element ay lies. Minor of an element aij is denoted by Mij.
Note: Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order (n – 1).
Cofactor
Cofactor of an element aij of a determinant, denoted by Aij or Cij is defined as Aij = (-1)i+j Mij, where Mij is a minor of an element aij.
Value of a Determinant
- Value of determinant of a matrix of order 2,
A = [Tex]\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} [/Tex]
[Tex]|A|=\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix} [/Tex]
[Tex]\Rightarrow |A|=a_{11}\cdot a_{22}-a_{21}\cdot a_{12} [/Tex]
- Value of determinant of a matrix of order 3,
[Tex]A = \begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix} [/Tex]
[Tex]|A|=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=a_{11}\cdot \begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}-a_{12}\cdot \begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}+ a_{13}\cdot \begin{vmatrix}a_{22}&a_{23}\\a_{31}&a_{32}\end{vmatrix} [/Tex]
Singular and Non-Singular Matrix
If the value of the determinant corresponding to a square matrix is zero, then the matrix is said to be a singular matrix, otherwise it is a non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it is said to be a non-singular matrix and of |A| = 0, then it is said to be a singular matrix.
Determinant Theorems
- If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
- The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.
Adjoint of a Matrix
The adjoint of a square matrix ‘A’ is the transpose of the matrix which is obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).
In general, the adjoint of a matrix A = [aij]n×n is a matrix [Aji]n×n, where Aji is a cofactor of element aji.
Properties of Adjoint of a Matrix
If A is a square matrix of order n × n, then
- A(adj A) = (adj A)A = |A| In
- |adj A| = |A|n-1
- adj (AT) = (adj A)T
The area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
[Tex]\Delta=\dfrac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix} [/Tex]
Inverse of a Square Matrix
Let A be a non-singular matrix such that |A| ≠ 0 then the inverse of the matrix is defined as
[Tex]A^{-1}=\dfrac{1}{|A|}adj(A) [/Tex]
Properties of an Inverse Matrix
- (A-1)-1 = A
- (AT)-1=(A-1)T
- (AB)-1 = B-1A-1
- (ABC)-1 =C-1B-1A-1
- adj (A-1) = (adj A)-1
Let the given system of equations be a1x + b1y + c1z = d1; a2x + b2y + c2z = d2 and a3x + b3y + c3z = d3.
Write the following system of linear equations in matrix form as
AX = B
where,
- [Tex]A=\begin{bmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix} [/Tex]
- [Tex]X=\begin{bmatrix}x\\y\\z\end{bmatrix} [/Tex]
- [Tex]B=\begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix} [/Tex].
Case I: If |A| ≠ 0, then the system is consistent and has a unique solution which is given by X = A-1B.
Case II: If |A| = 0 and (adj A) B ≠ 0, then the system is inconsistent and has no solution.
Case III: If |A| = 0 and (adj A) B = 0, then the system may be either consistent or inconsistent according to as the system has either infinitely many solutions or no solutions
Class 12 Maths Formulas
Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.
Table of Content
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability
- Chapter 6: Applications of Derivatives
- Chapter 7: Integrals
- Chapter 8: Applications of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
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