CBSE Class 12 Commerce Mathematics Determinants
CBSE Class 12 Commerce Mathematics Determinants:-we will provide complete details of CBSE Class 12 Commerce Mathematics Determinants in this article.CBSE Class 12 Commerce Mathematics Determinants
CBSE Class 12 Commerce Mathematics Determinants:-
In the previous chapter, we have studied about matrices and algebra of matrices. We have also learnt that a system of algebraic equations can be expressed in the form of matrices. This means, a system of linear equations like
a1 x + b1 y = c1
a2 x + b2 y = c2
can be represented as 1 1 1 2 2 2 a b x c a b y c ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ .
The number a1 b2 – a2 b1 which determines uniqueness of solution is associated with the matrix 1 1 2 2 A a b a b ⎡ ⎤ =⎢ ⎥ ⎣ ⎦ and is called the determinant of A or det A. Determinants have wide applications in Engineering, Science, Economics, Social Science, etc.
To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where aij = (i, j)th element of A.
This may be thought of as a function which associates each square matrix with a unique number (real or complex). If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k, where A ∈ M and k ∈ K, then f (A) is called the determinant of A. It is also denoted by |A| or det A or Δ.
If A = a b c d ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ , then determinant of A is written as |A| = a b c d = det (A)
CBSE Class 12 Commerce Mathematics Determinants:-Determinant of a matrix of order 3 × 3
Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and C3) giving the same value as shown below.
Consider the determinant of square matrix A = [aij]3 × 3
CBSE Class 12 Commerce Mathematics Determinants:-Expansion along first Row (R1)
Step 1 Multiply first element a11 of R1 by (–1)(1 + 1) [(–1)sum of suffixes in a11] and with the second order determinant obtained by deleting the elements of first row (R1) and first column (C1) of | A | as a11 lies in R1 and C1,
Step 2 Multiply 2nd element a12 of R1 by (–1)1 + 2 [(–1)sum of suffixes in a12] and the second order determinant obtained by deleting elements of first row (R1) and 2nd column (C2) of | A | as a12 lies in R1 and C2,
Step 3 Multiply third element a13 of R1 by (–1)1 + 3 [(–1)sum of suffixes in a13] and the second order determinant obtained by deleting elements of first row (R1) and third column (C3) of | A | as a13 lies in R1 and C3,
Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps 1, 2 and 3 above is given by
CBSE Class 12 Commerce Mathematics Determinants:-Properties of Determinants
In the previous section, we have learnt how to expand the determinants. In this section, we will study some properties of determinants which simplifies its evaluation by obtaining maximum number of zeros in a row or a column. These properties are true for determinants of any order. However, we shall restrict ourselves upto determinants of order 3 only.
Property 1 The value of the determinant remains unchanged if its rows and columns are interchanged.
Property 2 If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
Property 3 If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.
Proof If we interchange the identical rows (or columns) of the determinant Δ, then Δ does not change. However, by Property 2, it follows that Δ has changed its sign
Property 4 If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
Property 5 If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
Property 6 If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + k Cj .
Area of a Triangle
In earlier classes, we have studied that the area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3), is given by the expression 1/2 [x1(y2–y3) + x2 (y3–y1) +x3 (y1–y2)].
CBSE Class 12 Commerce Mathematics Determinants:-Minors and Cofactors
we will learn to write the expansion of a determinant in compact form using minors and cofactors.
Definition 1 Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.
Definition 2 Cofactor of an element aij , denoted by Aij is defined by Aij = (–1)i + j Mij , where Mij is minor of aij.
CBSE Class 12 Commerce Mathematics Determinants:-Adjoint and Inverse of a Matrix
In the previous chapter, we have studied inverse of a matrix. In this section, we shall discuss the condition for existence of inverse of a matrix. To find inverse of a matrix A, i.e., A–1 we shall first define adjoint of a matrix.
Adjoint of a matrix
Definition 3 The adjoint of a square matrix A = [aij]n × n is defined as the transpose of the matrix [Aij]n × n, where Aij is the cofactor of the element aij . Adjoint of the matrix A is denoted by adj A.
Definition 4 A square matrix A is said to be singular if A = 0.
Definition 5 A square matrix A is said to be non-singular if A ≠ 0
Theorem 2 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.
Theorem 3 The determinant of the product of matrices is equal to product of their respective determinants, that is, AB = A B , where A and B are square matrices of the same order
Theorem 4 A square matrix A is invertible if and only if A is non singular matrix.
CBSE Class 12 Commerce Mathematics Determinants:-Applications of Determinants and Matrices
we shall discuss application of determinants and matrices for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear equations.
Consistent system A system of equations is said to be consistent if its solution (one or more) exists.
Inconsistent system A system of equations is said to be inconsistent if its solution does not exist.
Solution of system of linear equations using inverse of a matrix
Let us express the system of linear equations as matrix equations and solve them using inverse of the coefficient matrix.
CBSE Class 12 Commerce Mathematics Determinants
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