Question

Applications of Determinants And Matrices

Answer 1

In this section, we shall discuss application of determinants and matrices for solving the system of linear equations in two or three variable and for checking the consistency of the system of linear equations.

Consistent system: A system of equation is said to be consistent if its solution (one or more) exists.

Inconsistent system : A system of equation is said to be inconsistent if its solution does not exist.

Solution of system of linear equation using inverse of a matrix

Let us express the system of linear equation as matrix equations and solve them using inverse of the matrix. Consider the system of equation

a₁x + b₁y + c₁z = d₁

a₂x + b₂ c₂z = d₂

a₃x + b₃y + c₃z = d₃

Case I : If A is non-singular matrix, then its inverse exists. Now.

or AX = B

or A^-1(AX) = A^-1B (Pre-multiplyingby A^-1)

or (A^-1A) X = A^-1B (By associative property)

or IX = A^-1B

or X = A^-1B.

This matrix equation provides unique solution for the given system of equation as inverse of matrix is unique. This method of solving system of equation is known as matrix method.

Case II: If A is a singular matrix, then |A| = 0.

In this case, we calculate (adj A) B

If (adj A) B ≠ 0 (0 being zero matrix), then solution does not exist and the solution of equation is called inconsistent.

If (adj A) B = 0, then system may be either consistent or inconsistent according as the system have either infinitely.

→ A determinant is defined as a (mapping) function from the set of square matrices to the set of real numbers.

→ Every square matrix A is associated with a number, called its determinant. Determinant is denoted by det (A) or |A| or A.

→ Expanding a determinant along any row or column gives same value. For easier calculations, we shall expand the determinant along that row or column which contains maximum number of zeros.

→ The value of the determinant remains unchanged if its rows and columns are interchanged.

→ If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

→ If any two rows (or columns) of a determinant are iddentical (all corresponding elements are same), then value of determinant is zero.

→ If each elemment of a row (or column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

→ If some or all elements of a row or column of a determinant are expressed as sum of two (or morre) terms, then the determinant can be exprressed as sum of two (or more) determinants.

→ If each element of a row (or column) of a determinant is zero, then its value is zero.

→ Let (x₁, y₁), (x₂, y₂), and (x₃, y₃) be the vertices of a triangle, then

Area of triangle = -[x₁(y₂ - y₃) + x₂ ( y₃ - y₁) + x₃ (y₁ - y₂)]

→ Area is a positive quantity, we always take the absolute value of the determinant.

→ The area of the triangle formed by three collinear points is zero.

→ Minor of an element of a_ij determinant is the determinant obtained by deleting its ith row and jth column in-which element a_ij lies.

→ Minor of an element a_ij is denoted by M_ij

→ Minor of an element of a determinant of order n (n > 2) is determinant of order n - 1.

→ If the minors are multiplied by the proper signs we get cofactors.

→ The cofactor of the element a_ij is A_ij (= (- 1)^1+jM_ij).

→ The signs to be multiplied are given by the rule.

→ Adjoint of a matrix is the transpose of the matrix formed by the cofactors of the given matrix.

→ A square matrix A is said to be singular if |A| =0.

→ A square matrix A is said to be non-singular if |A| ≠ 0.

→ A square matrix A is invertible if and only if A is non-singular matrix.

→ A system of equations is said to be consistent if its solution (one or more) exists.

→ A system of equations is said to be inconsistent if its solution does not exist.

→ Let the system of equations be :

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

For a square matrix A in matrix equation AX = B

• |A| ≠ 0, there exists unique solution
• |A| = 0 and (adj A) B ≠ 0, then there exists no solution.
• |A| = 0 and (adj A) B = p, then system may or may not be consistent.