Determinant Formula

Introduction

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

Note- Only square matrices have determinants.

Types of Determinants and their Solving

First order determinant

If A = [a], then det (A) = |A| = a

Second order determinant

If A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} , then

det (A) = |A| = a₁₁a₂₂ – a₂₁a₁₂

Third order determinant

If A = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}, then

det (A) = |A| = a₁₁(a₂₂a₃₃ – a₃₂a₂₃) – a₁₂(a₂₁a₃₃ – a₃₁a₂₃) + a₁₃(a₂₁a₃₂ – a₂₂a₃₁)

Evaluation of Determinant of Square Matrix of order 3 by Sarrus Rule

If A = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}, then determinant can be formed by enlarging the matrix by adjoining the first two columns of the right and draw lines as shown below parallel and perpendicular to the diagonal.

Δ = a₁₁a₂₂a₃₃ + a₁₂a₂₃a₃₁ + a₁₃a₂₁a₃₂ – a₁₃a₂₂a₃₁ – a₁₁a₂₃a₃₂ – a₁₂a₂₁a₃₃.

Note: This method doesn’t work for determinants of order greater than 3.

Properties of Determinants

(i) The value of the determinant remains unchanged, if rows are changed into columns and columns are changed into rows.

(ii) The interchange of any two rows (or columns) of the determinant changes its sign.

(iii) If all the elements of a row (or column) are zero, then the determinant is zero.

(iv) If any two rows (or columns) of a determinant are identical, then its value is zero.

(v) If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.

(vi) If each element of a row (or column) of a determinant is the sums of two or more terms, then the determinant can be expressed as the of two or more determinants

(vii) If the same multiple of the elements of any row (or column) of a determinant are added to the corresponding elements of any row (or column), then the value of the new determinant remains unchanged.

Minors and Cofactors

Minor of an element a_ij of a determinant is the determinant obtained by deleting its i^th row and j^th column in which element a_ij lies. Minor of an element a_ij is denoted by M_ij.

Remark: The minor of an element of a determinant of order n(n ≥ 2) is a determinant of order n – 1.

The cofactor of an element a_ij, denoted by A_ij is defined by A_ij = (–1)^{i + j} M_ij, where M_ij is minor of a_ij.

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

Area = ½ [x₁ (y₂ – y₃ ) + x₂ (y₃ –y₁ ) + x₃ (y₁ –y₂ )].

Now this expression can be written in the form of a determinant as

½{\begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1 \end{vmatrix} }

Remarks

1. Since area is a positive quantity, we always take the absolute value of the determinant in (1).
2. If the area is given, use both positive and negative values of the determinant for calculation.
3. The area of the triangle formed by three collinear points is zero.

Adjoint and Inverse of a Matrix

Adjoint of a matrix: The adjoint of a square matrix A = [a_ij] _{n × n} is defined as the transpose of the matrix [A_ij] _{n × n}, where A_ij is the cofactor of the element a_ij. Adjoint of the matrix A is denoted by adj A.

Let A = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}

Then adj A = Transpose of \begin{pmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23}\\ A_{31} & A_{32} & A_{33} \end{pmatrix}

Properties of the adjoint of a matrix

If A and B are two non-singular matrices of the same order n, then

1. A (adj A) = (adj A) A = |A| I
2. (adj A^T) = (adj A)^T
3. adj (AB) = adj B × adj A
4. adj (adj A) = |A|^n-2
5. |adj A| = |A|^n-1

Note : 

1. Adjoint of a diagonal matrix is a diagonal matrix.
2. Adjoint of a triangular Matrix is a triangular matrix
3. Adjoint of a symmetric matrix is a symmetric matric

Inverse of a Matrix : A^-1 = {1\over |A|}adj\ A

Properties of Inverse Matrix

Let A and B be two square matrices of the same order n. Then,

1. (A^-1)^-1 = A
2. (AB)^-1 = B^-1A^-1
3. (A^T)^-1 = (A^-1)^T
4. |A^-1| = |A|^-1
5. A.A^-1 = A^-1A = I

Singular and Non-Singular Matrix

Singular Matrix: A square matrix A is said to be singular if |A| = 0

Non-Singular Matrix: A square matrix A is said to be singular if |A| ≠ 0

1. Theorem: If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.
2. Theorem: The determinant of the product of matrices is equal to the product of their respective determinants, that is, AB = AB, where A and B are square matrices of the same order.
3. Theorem: A square matrix A is invertible if and only if A is a nonsingular matrix.

Remark

In general, if A is a square matrix of order n, then |adj(A)| = |A|^{n – 1}

Applications of Determinants and Matrices

1. Consistent system: A system of equations is said to be consistent if its solution (one or more) exists.
2. Inconsistent system: A system of equations is said to be inconsistent if its solution does not exist.

Solution of a system of linear equations using the inverse of a matrix

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

Consider the system of equations

a₁ x + b₁ y + c₁ z = d₁

a₂ x + b₂ y + c₂ z = d2

a₃ x + b₃ y + c₃ z = d₃

Let A = \begin{pmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{pmatrix}; X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} and B = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}

Case I – If A is a nonsingular matrix, then its inverse exists. Now

AX = B

or, A^–1 (AX) = A^–1 B (premultiplying by A^–1)

or, (A^–1A) X = A^–1 B (by the associative property)

or, IX = A^–1 B       [∵ A^–1A = I]

or, X = A^–1 B

This matrix equation provides a unique solution for the given system of equations as the inverse of a matrix is unique. This method of solving system of equations is known as the 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 ≠ O, (O being zero matrices), then the solution does not exist and the system of equations is called inconsistent.
• If (adj A) B = O, then the system may be either consistent or inconsistent according to the system has either infinitely many solutions or no solution

Solution of a system of linear equations using the Cramer’s Rule

Consider the system of equations

a₁ x + b₁ y + c₁ z = d₁

a₂ x + b₂ y + c₂ z = d2

a₃ x + b₃ y + c₃ z = d₃

Then the solution of the system of equations is

x = Δ_x \over Δ; y = Δ_y \over Δ; z = Δ_z \over Δ

Where, Δ = {\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix} }

Δ_{x =} {\begin{vmatrix} d_{1} & b_{1} & c_{1}\\ d_{2} & b_{2} & c_{2}\\ d_{3} & b_{3} & c_{3} \end{vmatrix} }

Δ_y = {\begin{vmatrix} a_{1} & d_{1} & c_{1}\\ a_{2} & d_{2} & c_{2}\\ a_{3} & d_{3} & c_{3} \end{vmatrix} }

Δ_z = {\begin{vmatrix} a_{1} & b_{1} & d_{1}\\ a_{2} & b_{2} & d_{2}\\ a_{3} & b_{3} & d_{3} \end{vmatrix} }

1. If D ≠ 0, then the system of equation is consistent with the unique solution.
2. If D = 0 and at least one of the determinants D₁, D₂, D₃ is non-zero, then the given system is inconsistent, i.e. having no solution.
3. If D = 0 and D₁ = D₂ = D₃ = 0, then the system is consistent, with infinitely many solutions.
4. If D ≠ 0 and D₁ = D₂ = D₃ = 0, then systemhas only trivial solution, (x = y = z = 0).