Determinants of a Matrix

Determinant is a scalar value that can be calculated from the elements of a square matrix.

\(\begin{array}{l}\text{It  is an arrangement of numbers in the form}\ \left| \begin{matrix} a & b \\ c & d \\ \end{matrix} \right|.\end{array} \)
Determinant for a 3×3 matrix is determined by;
\(\begin{array}{l}\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2}& b_{2} & c_{2}\\ a_{3}& b_{3} & c_{3} \end{vmatrix}\end{array} \)
= a1(b2c3 – b3c2) – b1(a2c3 – a3c2) + c1(a2b3 – a3b2). In this article, we come across properties of determinants, multiplication of determinants and determinants formula. 

Introduction to Determinants

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

\(\begin{array}{l}E.g.\left. \begin{matrix} {{a}_{1}}x+{{b}_{1}}y={{c}_{1}} \\ {{a}_{2}}x+{{b}_{2}}y={{c}_{2}} \\ \end{matrix} \right] \Rightarrow x=\frac{{{b}_{2}}{{c}_{1}}-{{b}_{1}}{{c}_{2}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\;\;\; and \;\; y=\frac{{{a}_{1}}{{c}_{2}}-{{a}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\end{array} \)

Mathematicians defined the symbol

\(\begin{array}{l}\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{b}_{2}} \\ \end{matrix} \right|\end{array} \)
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 
\(\begin{array}{l}\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{b}_{2}} \\ \end{matrix} \right|\end{array} \)
 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.

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

For every square matrix A of order n 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,

\(\begin{array}{l}A=\begin{bmatrix} {{a}} & {{b}} \\ {{c}} & {{d}} \\ \end{bmatrix},\ \text{the determinant is ad – bc.}\end{array} \)
\(\begin{array}{l}\text{In the case of a 3 x 3 matrix}\ A=\begin{bmatrix} a &b &c \\ d &e &f \\ g&h &i \end{bmatrix},\end{array} \)
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 the 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

Illustration 1: Expand

\(\begin{array}{l}\left| \begin{matrix} 3 & 2 & 5 \\ 9 & -1 & 4 \\ 2 & 3 & -5 \\ \end{matrix} \right|\end{array} \)

by Sarrus rules.

Solution:

By using Sarrus rule i.e.

\(\begin{array}{l}\Delta =\left( {{a}_{11}}{{a}_{22}}{{a}_{33}}+{{a}_{12}}{{a}_{23}}{{a}_{31}}+{{a}_{13}}{{a}_{21}}{{a}_{32}} \right)-\left( {{a}_{13}}{{a}_{22}}{{a}_{31}}+{{a}_{11}}{{a}_{23}}{{a}_{32}}+{{a}_{12}}{{a}_{21}}{{a}_{33}} \right)\end{array} \)
we can expand the given determinant.

Determinant Example Question

 

Here,

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

Illustration 2: Evaluate the determinant :

\(\begin{array}{l}\left| \begin{matrix} {{x}^{2}}-x+1 & x-1 \\ x+1 & x+1 \\ \end{matrix} \right|\end{array} \)

Solution:

By using determinant expansion formula we can get the result.

We have

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

Symmetric and Skew Symmetric Determinants

Symmetric Determinant

A determinant is called a Symmetric Determinant if aij = aji, ∀ i, j

\(\begin{array}{l}e.g.\; [\left| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix} \right|\end{array} \)

Skew Symmetric Determinant

A determinant is called a skew symmetric determinant if aij = -aji, ∀ i, j for every element.

\(\begin{array}{l}E.g.\ \left| \begin{matrix} 0 & 3 & -1 \\ -3 & 0 & 5 \\ 1 & -5 & 0 \\ \end{matrix} \right|\end{array} \)

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)

\(\begin{array}{l}\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{b}_{2}} \\ \end{matrix} \right|\times \left| \begin{matrix} {{l}_{1}} & {{m}_{1}} \\ {{l}_{2}} & {{m}_{2}} \\ \end{matrix} \right|=\left| \begin{matrix} {{a}_{1}}{{l}_{1}}+{{b}_{1}}{{l}_{2}} & {{a}_{1}}{{m}_{1}}+{{b}_{1}}{{m}_{2}} \\ {{a}_{2}}{{l}_{1}}+{{b}_{2}}{{l}_{2}} & {{a}_{2}}{{m}_{1}}+{{b}_{2}}{{m}_{2}} \\ \end{matrix} \right|\end{array} \)

(b) Multiplication of two third order determinants is defined

\(\begin{array}{l}\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|\times \left| \begin{matrix} {{l}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}} \\ {{l}_{3}} & {{m}_{3}} & {{n}_{3}} \\ \end{matrix} \right|\end{array} \)

(as R to C method)

\(\begin{array}{l}=\left| \begin{matrix} {{a}_{1}}{{l}_{1}}+{{b}_{1}}{{l}_{2}}+{{c}_{1}}{{l}_{3}} & {{a}_{1}}{{m}_{1}}+{{b}_{1}}{{m}_{2}}+{{c}_{1}}{{m}_{3}} & {{a}_{1}}{{n}_{1}}+{{b}_{1}}{{n}_{2}}+{{c}_{1}}{{n}_{3}} \\ {{a}_{2}}{{l}_{1}}+{{b}_{2}}{{l}_{2}}+{{c}_{2}}{{l}_{3}} & {{a}_{2}}{{m}_{1}}+{{b}_{2}}{{m}_{2}}+{{c}_{2}}{{m}_{3}} & {{a}_{2}}{{n}_{1}}+{{b}_{2}}{{n}_{2}}+{{c}_{2}}{{n}_{3}} \\ {{a}_{3}}{{l}_{1}}+{{b}_{3}}{{l}_{2}}+{{c}_{3}}{{l}_{3}} & {{a}_{3}}{{m}_{1}}+{{b}_{3}}{{m}_{2}}+{{c}_{3}}{{m}_{3}} & {{a}_{3}}{{n}_{1}}+{{b}_{3}}{{n}_{2}}+{{c}_{3}}{{n}_{3}} \\ \end{matrix} \right|\end{array} \)

Note:

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

(ii) To get the Tmn (term in the mth row nth column) in the product, Take the mth row of the 1st determinant and multiply it by the corresponding terms of the nth column of the 2nd determinant and add.

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

(iv) IfΔ′ is the determinant formed by replacing the elements of a Δ of order n by their corresponding co-factors then

Δ’ = Δn-1 . Δ’ is called the reciprocal determinant.

Illustration 3: Reduce the power of the determinant

\(\begin{array}{l}{{\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|}^{2}}\end{array} \)

to 1.

Solution:

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

\(\begin{array}{l}{{\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|}^{2}}=\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|\Rightarrow \left| \begin{matrix} {{b}^{2}}+{{c}^{2}} & ab & ac \\ ba & {{c}^{2}}+{{a}^{2}} & bc \\ ca & cb & {{a}^{2}}+{{b}^{2}} \\ \end{matrix} \right|\end{array} \)

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Frequently Asked Questions

What is the determinant of an identity matrix?

The determinant of an identity matrix is 1.

Are Determinants always positive?

No. Determinants can be negative, positive or zero.

Give an application of determinants.

Determinants are used to solve linear equations in two or three variables.

What is the value of the determinant of a matrix, if any two rows or columns are identical?

If any two rows or columns of a matrix are identical, then the value of the determinant is equal to zero.

What will be the determinant of a matrix when two rows and columns are interchanged?

If two rows or columns of a determinant are interchanged, then the determinant changes its sign.

Will the determinant change, if transposed?

No. The determinant of a matrix is equal to the determinant of transpose of the matrix. |AT| = |A|.

Does a matrix have more than one determinant?

No. A matrix cannot have more than one determinant.

What happens to a determinant, if all the elements of a row/column are multiplied by any scalar k?

If all elements of a row/column of a determinant are multiplied by some scalar k, the value of the new determinant is k times the given determinant. If A is a square matrix having n rows, and k be any scalar, then |kA| = kn|A|.

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