Determinant of a Matrix

Introduction to Determinants

Development of determinants took place when mathematicians were trying to solve a system of simultaneous linear equations.

Mathematicians defined the symbol $\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{b}_{2}} \\ \end{matrix} \right|$ as a determinant of order 2 and the four numbers arranged in row and column were called its elements. If we write the coefficients of the equations in the following form $\left| \begin{matrix} a & b \\ c & d \\ \end{matrix} \right|$ then such an arrangement is called a determinant. In a determinant, horizontal lines are known as rows and vertical lines are known as columns. The shape of every determinant is a square. If a determinant is of order n then it contains n rows and n columns.

E.g. $\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{b}_{2}} \\ \end{matrix} \right|,\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|$ are determinants of second and third order respectively.

For every square matrix A of order m x n, there exists a number associated with it called the determinant of a square matrix.

For a matrix of 1 x 1, the determinant is A = [a].

For a 2 x 2 matrix, $A=\left| \begin{matrix} {{a}} & {{b}} \\ {{c}} & {{d}} \\ \end{matrix} \right|$ the determinant is ad – bc.

In the case of a 3 x 3 matrix $A=\begin{bmatrix} a &b &c \\ d &e &f \\ g&h &i \end{bmatrix}$, the value of determinant is = a (ei − fh) − b (di − fg) + c (dh − eg).

Note:

(i) The number of elements in a determinant of order n is n².

(ii) A determinant of order 1 is the number itself.

Properties of Determinants

• There will be no change in the value of determinant if the rows and columns are interchanged.
• Suppose any two rows or columns of a determinant are interchanged, then its sign changes.
• If any two rows or columns of a determinant are the same, then the determinant is 0.
• If any row or column of the determinant is multiplied by a variable k, then its value is multiplied by k.
• Say if some or all elements of a row or column are expressed as the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.

Contents in Determinants

• Introduction to Determinants
• Minors and Cofactors
• Properties of Determinants in detail
• System of Linear Equations using Determinants
• Differentiation and Integration of Determinants
• Standard Determinants

Illustration 1: Expand $\left| \begin{matrix} 3 & 2 & 5 \\ 9 & -1 & 4 \\ 2 & 3 & -5 \\ \end{matrix} \right|$

by Sarrus rules.

Solution:

By using Sarrus rule i.e. $\Delta =\left( {{a}_{11}}{{a}_{22}}{{a}_{33}}+{{a}_{12}}{{a}_{23}}{{a}_{31}}+{{a}_{13}}{{a}_{21}}{{a}_{32}} \right)-\left( {{a}_{13}}{{a}_{22}}{{a}_{31}}+{{a}_{11}}{{a}_{23}}{{a}_{32}}+{{a}_{12}}{{a}_{21}}{{a}_{33}} \right)$we can expand the given determinant.

Here, $\Delta =\left| \begin{matrix} 3 & 2 & 5 \\ 9 & -1 & 4 \\ 2 & 3 & -5 \\ \end{matrix} \right|\Rightarrow \Delta =15-36+90+16+135+10=230$

Illustration 2: Evaluate the determinant : $\left| \begin{matrix} {{x}^{2}}-x+1 & x-1 \\ x+1 & x+1 \\ \end{matrix} \right|$

Solution:

By using determinant expansion formula we can get the result.

We have $\left| \begin{matrix} {{x}^{2}}-x+1 & x-1 \\ x+1 & x+1 \\ \end{matrix} \right|=\left( {{x}^{2}}-x+1 \right)\left( x+1 \right)-\left( x+1 \right)\left( x-1 \right)={{x}^{3}}+{{x}^{2}}-{{x}^{2}}-x+x+1-{{x}^{2}}+1={{x}^{3}}-{{x}^{2}}+2$

Symmeteric and Skew Symmetric Determinants

Symmetric Determinant

A determinant is called a Symmetric Determinant if ${{a}_{i\,j}}={{a}_{j\,i}},\forall \,i,j \;e.g.\; [\left| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix} \right|$

Skew Symmetric Determinant

A determinant is called a skew symmetric determinant if ${{a}_{i\,j}}=-{{a}_{j\,i}}\forall i,\,j$ for every element.

E.g. $\left| \begin{matrix} 0 & 3 & -1 \\ -3 & 0 & 5 \\ 1 & -5 & 0 \\ \end{matrix} \right|$

Note: (i) det |A| = 0 ⇒A is singular matrix (ii) det | A | ≠ 0 ⇒A is non-singular matrix

Multiplication of Two Determinants

(a) Multiplication of two second order determinants as follows: (as R to C method)

(b) Multiplication of two third order determinants is defined

(as R to C method)

Note:

(i) The two determinants to be multiplied must be of the same order.

(ii) To get the ${{T}_{mn}}$ (term in the m^th row n^th column) in the product, Take the m^th row of the 1^st determinant and multiply it by the corresponding terms of the n^th column of the 2^nd determinant and add.

(iii) This method is the row by column multiplication rule for the product of 2 determinants of the n^rd order determinant.

(iv) IfΔ′ is the determinant formed by replacing the elements of a Δ of order n by their corresponding co-factors then $\Delta ‘={{\Delta }^{n-1}}.$ $(\Delta ‘)$ is called the reciprocal determinant).

Illustration 3: Reduce the power of the determinant ${{\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|}^{2}}$

to 1.

Solution:

By multiplying the given determinant two times we get the determinant as required.

Determinant of cofactors of D is

(Row interchanging into columns) $= {{\left| \begin{matrix} x & c & -b \\ -c & x & a \\ b & -a & x \\ \end{matrix} \right|}^{2}} \;\;({{D}^{C}}={{D}^{2}},D \;is \;third\; order determinant)$

Hence $\left| \begin{matrix} {{a}^{2}}+{{x}^{2}} & ab-cx & ac+bx \\ ab+cx & {{b}^{2}}+{{x}^{2}} & bc-ax \\ ac-bx & bc+ax & {{c}^{2}}+{{x}^{2}} \\ \end{matrix} \right|={{\left| \begin{matrix} x & c & -b \\ -c & x & a \\ b & -a & x \\ \end{matrix} \right|}^{2}}$