Determinants

Determinants are a fascinating concept in linear algebra that helps us understand and solve matrix equations. Imagine you have a set of linear equations, and you want to find if they have a unique solution. Determinants give you a quick way to check this. They are like a special number that can tell us a lot about a matrix, such as whether it has an inverse or if the system of equations it represents has a unique solution. In essence, determinants are powerful tools that simplify complex matrix operations and provide deep insights into the structure of linear systems.

In this article, you'll discover practical tips and tricks for using determinants to solve matrix problems easily. With clear examples and illustrations, we'll help you tackle determinant challenges confidently. 

1.0Determinants Definition

It is the value associated with the square arrangement of numbers in rows and columns written in between two parallel lines.

For every square matrix A = a_ij of order n, we can assign a number (real or complex) known as the determinant of the matrix A. Here, a_ij represents the element in the i-th row and j-th column of A. This determinant is a special value that provides critical information about the matrix, such as whether it can be inverted, the volume scaling factor of the linear transformation it represents, and whether a system of linear equations possesses a unique solution.

Ex- $\left|\begin{array}{ll}{a}_1& b_1\\ a_2& b_2\end{array}\right|$

A determinant of second order consists of two rows and two columns. 

$\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

A determinant of third order or 3 × 3 consists of three rows and three columns. 

2.0Value of a Determinant

_D=$\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=a_1\left|\begin{array}{ll}{b}_2& c_2\\ b_3& c_3\end{array}\right|-b_1\left|\begin{array}{ll}{a}_2& c_2\\ a_3& c_3\end{array}\right|+c_1\left|\begin{array}{ll}{a}_2& b_2\\ a_3& b_3\end{array}\right|$

= a₁(b₂c₃ – b₃c₂) – b₁(a₂c₃ – a₃c₂) + c₁(a₂b₃ – a₃b₂)

3.0By SARRUS Method

_D=$\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=$

(a₁b₂c₃ + a₂b₃c₁ + a₃b₁c₂) – (a₃b₂c₁ + a₂b₁c₃ + a₁b₃c₂) 

4.0Minors 

 Minors of an element is defined as the minor determinant obtained by deleting a particular row or column in which that element lies. Ex. in the determinant

_D=$\left|\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|\text{ minor of }{a}_{12}\text{ denotes as }{M}_{12}$

${\mathbf{M}}_{12}=\left|\begin{array}{ll}{\mathbf{a}}_{21}& {\mathbf{a}}_{23}\\ {\mathbf{a}}_{31}& {\mathbf{a}}_{33}\end{array}\right|\text{ and so on }$

5.0Cofactor

Cofactor of an elements a_ij is expressed as C_ij and is calculated as:

C_ij = (–1) ^i+j M_ij, Where ‘i’ denotes the row and j denotes the column.

Note: That a determinant of order 3 will have 9 minors, each minor will be a determinant of order 2 and a determinant of order4 will have 16 minors each minor will be a determinant of order 3.

Expansion of a determinant in Terms of the Elements of any Row or column

$\mathbf{D}={\sum }_{i=1}^3{a}_{ij}{C}_{ij}\text{ etc. for }\mathbf{i}=1,2,3$

Sum of the product of elements of any row (column) with cofactors of corresponding elements of any other row (column) is ZERO.

6.0Properties of Determinants

• Property 1: The value of a determinant stays the same if the respective row and columns are interchanged.
• Property 2: If any 2 rows (or columns) of a determinant are interchanged, the value of determinant is changed in sign only.
• Property 3: If a determinant has any two identical rows (or columns), its value is zero.

$\mathrm{D}=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_1& b_1& c_1\\ a_3& b_3& c_3\end{array}\right|=$ 0

• Property 4: If all the elements of a row (or column) are multiplied by a scalar, the determinant is multiplied by that scalar. 

$\text{ If }\mathrm{D}=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\text{ and }{\mathrm{D}}_1=\left|\begin{array}{ccc}K{a}_1& K{b}_1& K{c}_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

then  D₁ = KD

Note: |K D|=K^n .|D|  where n is the order of determinants and K is a scalar.

• Property 5: If each element of a row (or column) can be written as the sum of two terms, the determinant can be expressed as the sum of two determinants.

If

$\left|\begin{array}{ccc}{\mathbf{a}}_1+\mathbf{x}& {\mathbf{b}}_1+\mathbf{y}& {\mathbf{c}}_1+\mathbf{z}\\ {\mathbf{a}}_2& {\mathbf{b}}_2& {\mathbf{c}}_2\\ {\mathbf{a}}_3& {\mathbf{b}}_3& {\mathbf{c}}_3\end{array}\right|=\left|\begin{array}{ccc}{\mathbf{a}}_1& {\mathbf{b}}_1& {\mathbf{c}}_1\\ {\mathbf{a}}_2& {\mathbf{b}}_2& {\mathbf{c}}_2\\ {\mathbf{a}}_3& {\mathbf{b}}_3& {\mathbf{c}}_3\end{array}\right|+\left|\begin{array}{ccc}\mathbf{x}& \mathbf{y}& \mathbf{z}\\ {\mathbf{a}}_2& {\mathbf{b}}_2& {\mathbf{c}}_2\\ {\mathbf{a}}_3& {\mathbf{b}}_3& {\mathbf{c}}_3\end{array}\right|$

• Property 6: The value of a determinant is not changed if a multiple of one row (or column) is added to another row (or column).

$\text{ Let }D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\text{ and }{D}^{\prime }=\left|\begin{array}{ccc}{a}_1+m{a}_2& b_1+\mathbf{m}{b}_2& c_1+m{c}_2\\ a_2& b_2& c_2\\ a_3+n{a}_1& b_3+n{b}_1& c_3+\mathbf{n}{c}_1\end{array}\right|$

These properties simplify the manipulation and calculation of determinants, making it easier to understand and solve complex matrix equations.

7.0Multiplication of Two Determinants

If A and B  are two square matrices of same order, then $|AB|=|A||B|$

• Multiplication of two determinants of order 2 × 2

$\left|\begin{array}{ll}{a}_1& b_1\\ a_2& b_2\end{array}\right|\times \left|\begin{array}{ll}{\ell }_1& m_1\\ {\ell }_2& m_2\end{array}\right|=\left|\begin{array}{ll}{a}_1{\ell }_1+b_1{\ell }_2& a_1{m}_1+b_1{m}_2\\ a_2{\ell }_1+b_2{\ell }_2& a_2{m}_1+b_2{m}_2\end{array}\right|$

• Multiplication of two determinants of order 3 × 3

$\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\times \left|\begin{array}{lll}{\ell }_1& m_1& n_1\\ {\ell }_2& m_2& n_2\\ {\ell }_3& m_3& n_3\end{array}\right|=\left|\begin{array}{lll}{a}_1{\ell }_1+b_1{\ell }_2+c_1{\ell }_3& a_1{m}_1+b_1{m}_2+c_1{m}_3& a_1{n}_1+b_1{n}_2+c_1{n}_3\\ a_2{\ell }_1+b_2{\ell }_2+c_2{\ell }_3& a_2{m}_1+b_2{m}_2+c_2{m}_3& a_2{n}_1+b_2{n}_2+c_2{n}_3\\ a_3{\ell }_1+b_3{\ell }_2+c_3{\ell }_3& a_3{m}_1+b_3{m}_2+c_3{m}_3& a_3{n}_1+b_3{n}_2+c_3{n}_3\end{array}\right|$

Note: Multiplication of 2 matrices is only possible with Row - column  Method, while multiplication of 2 determinants can be done with R-R , C-C, C-R and R-R method also.

8.0Solved Examples of Determinants

Example 1: The value of $\left|\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\end{array}\right|$ is-  

Solution: 

$\left|\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\end{array}\right|=1\left|\begin{array}{cc}3& 6\\ -7& 9\end{array}\right|-2\left|\begin{array}{cc}-4& 6\\ 2& 9\end{array}\right|+3\left|\begin{array}{cc}-4& 3\\ 2& -7\end{array}\right|$

= (27 + 42) – 2 (–36 – 12) + 3 (28 – 6) = 231 

Example 2: Find the minors and cofactors of elements '-3', '5', '–1' and '7' in the determinant $\left|\begin{array}{ccc}2& -3& 1\\ 4& 0& 5\\ -1& 6& 7\end{array}\right|.$

Solution:

$\text{ Minor of }-3=\left|\begin{array}{cc}4& 5\\ -1& 7\end{array}\right|=33\text{; Cofactor of }-3=-33\text{ Minor of }5=\left|\begin{array}{cc}2& -3\\ -1& 6\end{array}\right|=9;\text{ Cofactor of }5=-9\text{ Minor of }-1=\left|\begin{array}{cc}-3& 1\\ 0& 5\end{array}\right|=-15;\text{ Cofactor of }-1=-15\text{ Minor of }7=\left|\begin{array}{cc}2& -3\\ 4& 0\end{array}\right|=12;\text{ Cofactor of }7=12$

Example 3: If  $D_r=\left|\begin{array}{ccc}r& r^3& 2\\ n& n^3& 2n\\ \frac{n(n+1)}{2}& {\left(\frac{n(n+1)}{2}\right)}^2& 2(n+1)\end{array}\right|\text{, find }{\sum }_{r=0}^n{D}_r,find{\sum }_{r=0}^n{D}_r.$

Solution:

${\sum }_{r=0}^n{D}_r=\left|\begin{array}{ccc}{\sum }_{r=0}^{n}r& {\sum }_{r=0}^n{r}^3& {\sum }_{r=0}^{n}2\\ n& n^3& 2n\\ \frac{n(n+1)}{2}& {\left(\frac{n(n+1)}{2}\right)}^2& 2(n+1)\end{array}\right|=\left|\begin{array}{ccc}\frac{n(n+1)}{2}& {\left(\frac{n(n+1)}{2}\right)}^2& 2(n+1)\\ n& n^3& 2n\\ \frac{n(n+1)}{2}& {\left(\frac{n(n+1)}{2}\right)}^2& 2(n+1)\end{array}\right|=0$

Example 4:$\text{ If }\left|\begin{array}{ccc}{3}^2+k& 4^2& 3^2+3+k\\ 4^2+k& 5^2& 4^2+4+k\\ 5^2+k& 6^2& 5^2+5+k\end{array}\right|=0\text{, then the value of }k\text{ is }$ ?

Solution: 

$\mathrm{Applying}\left({}^{C_3}\to C_3-C_1\right)D=\left|\begin{array}{ccc}{3}^2+k& 4^2& 3\\ 4^2+k& 5^2& 4\\ 5^2+k& 6^2& 5\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}9+k& 16& 3\\ 7& 9& 1\\ 9& 11& 1\end{array}\right|=0\left({\mathrm{R}}_3\to {\mathrm{R}}_3-{\mathrm{R}}_2;{\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_1\right)$

⇒    k – 1 = 0 ⇒ k =1

Example 5: Show that $\left|\begin{array}{lll}{a}^2+l^2& ab+l& ca-l\\ ab-l& b^2+l^2& bc+l\\ ac+l& bc-l& c^2+l^2\end{array}\right|\times \left|\begin{array}{ccc}l& c& -b\\ -c& l& a\\ b& -a& l\end{array}\right|=l^3{\left(l^2+a^2+b^2+c^2\right)}^3$

Solution:

We observe that the elements in the first determinant are the cofactors of corresponding elements of the second determinant.

$\text{ So }\left|\begin{array}{ccc}\lambda & c& -b\\ -c& \lambda & a\\ b& -a& \lambda \end{array}\right|={\left[\lambda \left({\lambda }^2+a^2+b^2+c^2\right)\right]}^3$

= λ³(λ² + a² + b² + c²)³

Example 6: How do you calculate the determinant of a 2x2 matrix?

Solution:

For a 2×2 matrix $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$

the determinant is calculated as:

det(A) = ad – bc

Example 7: How do you find the determinant of a 3x3 matrix?

Solution:

For a 3x3 matrix $A=\left[\begin{array}{lll}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$, the determinant can be calculated using the rule of Sarrus or cofactor expansion:

det(A) = a(ei - fh) – b(di – fg) + c(dh – eg)

9.0Practice Questions on Determinants

1. Let A=$\left[\begin{array}{lll}{\cos }^{-1}x& {\cos }^{-1}y& {\cos }^{-1}z\\ {\cos }^{-1}y& {\cos }^{-1}z& {\cos }^{-1}x\\ {\cos }^{-1}z& {\cos }^{-1}x& {\cos }^{-1}y\end{array}\right]$

such that |A| = 0, then maximum value of  x + y +z is 

(A) 3 (B) 0 (C) 1 (D) 2

2. If α, β & γ are the roots of the equation x³ + px + q = 0 , then the value of the determinant =

(A) p (B) q (C) p² – 2q (D) none

3. $\text{ The determinant }\left|\begin{array}{ccc}\cos (\theta +\varphi )& -\sin (\theta +\varphi )& \cos 2\varphi \\ \sin \theta & \cos \theta & \sin \varphi \\ -\cos \theta & \sin \theta & \cos \varphi \end{array}\right|$ is:

(A) 0 (B) independent of θ

(C) independent of φ (D) independent of θ & φ both

4. Value of the  =$\left|\begin{array}{lll}{a}^3-x& a^4-x& a^5-x\\ a^5-x& a^6-x& a^7-x\\ a^7-x& a^8-x& a^9-x\end{array}\right|$ is

(A) 0 (B) ( – 1) ( – 1) ( – 1)

(C) ( + 1) ( + 1) ( + 1) (D) – 1