Determinant of a Matrix

The determinant of a matrix is an important number in math that tells us if a matrix can be inverted. For 2×2, 3×3, and 4×4 matrices, there are specific ways to calculate it. The determinant helps solve linear equations and understand properties like eigenvalues and shapes in geometry. Learn how to find and use the determinant of a matrix easily. 

1.0Definition of Determinant of Matrix

The determinant of a matrix is a single numerical value that provides important insights into the properties and behavior of the matrix and the linear transformation it represents. For a square matrix (a matrix with the equal no of rows and columns), the determinant can indicate whether the matrix is invertible (i.e., if it has an inverse). A non-zero determinant indicates that the matrix is invertible, while a zero determinant means the matrix is singular and non-invertible.

• Determinant for a 2 × 2 matrix is determined

$\left|\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right|=\mathrm{ad}-\mathrm{bc}$

• Determinant for a 3 × 3 matrix is determined by

$\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ c_3& c_3& c_3\end{array}\right|=a_1\left(b_2{c}_3-b_3{c}_2\right)-b_1\left(a_2{c}_3-a_3{c}_2\right)+c_1\left(a_2{b}_3-a_3{b}_2\right)$

2.0Matrix Minors of a Determinants

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

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

3.0Cofactors of a Determinants of Matrix

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:

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

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

4.0Determinant Matrix Properties

Property 1: The value of a determinant stays the same if the row and columns are interchanged.

Property 2: If any 2 rows (or columns) of a matrix are interchanged, the determinant's value changes sign.

Property 3: If a determinant has any two identical rows (or columns), its value is zero.

If  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. 

If $\mathrm{D}=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|{\mathrm{D}}^{‘}=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

 then $D^{\prime }=D{D}^{\prime }=D$

Note: $|KD|=K^n.|D|$ where n is the order of determinants.

Property 5: If each element of a row (or column) is written as the sum of two terms, the determinant can be written as the sum of 2 separate determinants.

If $\left|\begin{array}{ccc}{a}_1+\mathbf{x}& {\mathbf{b}}_1+\mathbf{y}& c_1+\mathbf{z}\\ {\mathbf{a}}_2& {\mathbf{b}}_2& {\mathbf{c}}_2\\ {\mathbf{a}}_3& {\mathbf{b}}_3& {\mathbf{c}}_3\end{array}\right|=\left|\begin{array}{ccc}{\mathbf{a}}_1& {\mathbf{b}}_1& {\mathbf{c}}_1\\ {\mathbf{a}}_2& {\mathbf{b}}_2& {\mathbf{c}}_2\\ {\mathbf{a}}_3& {\mathbf{b}}_3& {\mathbf{c}}_3\end{array}\right|+\left|\begin{array}{ccc}\mathbf{x}& \mathbf{y}& \mathbf{z}\\ {\mathbf{a}}_2& {\mathbf{b}}_2& c_2\\ {\mathbf{a}}_3& {\mathbf{b}}_3& {\mathbf{c}}_3\end{array}\right|$

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).

$\text{ Let }D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\text{ and }{D}^{\prime }=\left|\begin{array}{ccc}{a}_1+m{a}_2& b_1+\mathbf{m}{\mathbf{b}}_2& {\mathbf{c}}_1+\mathbf{m}{\mathbf{c}}_2\\ {\mathbf{a}}_2& {\mathbf{b}}_2& {\mathbf{c}}_2\\ {\mathbf{a}}_3+\mathbf{n}{\mathbf{a}}_1& {\mathbf{b}}_3+\mathbf{n}{\mathbf{b}}_1& {\mathbf{c}}_3+\mathbf{n}{\mathbf{c}}_1\end{array}\right|$

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

5.0Determinants of Matrix Formula (SARRUS Method)

By 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) 

6.0Determinant of Square matrix

The determinant of a square matrix is a scalar quantity that reveals key information about the matrix and the linear transformation it embodies. For a matrix to be square, it must have the same number of rows and columns, such as 2 × 2, 3 × 3, 4 × 4, or in general, n × n. 

Calculation Methods

For an n × n matrix A, the determinant, denoted as det(A) or $|\mathrm{A}|$ can be calculated using various methods:

1. Cofactor Expansion: 

• This method involves expanding along any row or column using cofactors:

$\det (A)={\sum }_{j=1}^n(-1{)}^{i+j}\cdot a_{ij}\cdot \det \left(M_{ij}\right)$

• Here, M_ij is the minor of element a_ij, calculated as the determinant of the (n – 1) × (n – 1) matrix obtained by removing the i^th row and i^th column from A.

2. Triangularization: 

• Transforming A into an upper or lower triangular matrix U or L, where the determinant is the product of the diagonal elements:

$\det (A)={\prod }_{i=1}^n{u}_{ii}\text{ or }\det (A)={\prod }_{i=1}^n{l}_{ii}$

3. Eigenvalues: 

If A has eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ , then:

$\det (A)={\lambda }_1\cdot {\lambda }_2\cdot \dots \cdot {\lambda }_n$

7.0Determinant of non-square matrix.

The determinant is a concept specifically defined for square matrices, meaning matrices with an equal number of rows and columns. For non-square matrices, the determinant is not defined. This is because many properties and operations that rely on the determinant, such as checking matrix invertibility or solving systems of linear equations using Cramer's rule, are inherently tied to the concept of a square matrix.

Key Points:

1. Non-Square Matrices: These matrices have a different number of rows and columns (e.g., m × n where m ≠ n). 
2. Undefined Determinant: For non-square matrices, the determinant is not defined, as the fundamental properties and applications of the determinant do not apply to these matrices.

8.0Cramer’s Rule (System of Linear Equation)

(a) Equations involving two variables:

(i) Consistent Equations : Definite & unique solution (Intersecting lines)

(ii) Inconsistent Equations : No solution (Parallel lines)

(iii) Dependent Equations : Infinite solutions (Identical lines)

Let, a₁ x + b₁ y + c₁ = 0 

a₂ x + b₂ y + c₂ = 0 then:

(1) $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}\Rightarrow$ Given equations are consistent with unique solution

(2) $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}\Rightarrow$ ⇒ Given equations are inconsistent

(3) $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\Rightarrow$ Given equations are consistent with infinite solutions

(b) Equations Involving Three variables:

Let a₁ x + b₁ y + c₁ = d₁ ...(i)

a₂ x + b₂ y + c₂ = d₂ ...(ii) 

a₃ x + b₃ y + c₃ = d₃ ...(iii)

Then, x = $\frac{D_1}{D},\mathrm{y}=\frac{D_2}{D},\mathrm{z}=\frac{D_3}{D}$ .

Where

$\mathrm{D}=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|;{\mathrm{D}}_1=\left|\begin{array}{lll}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|;$

${\mathrm{D}}_2=\left|\begin{array}{lll}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|\&{\mathrm{D}}_3=\left|\begin{array}{lll}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|$ 

Note: (i) If D ≠ 0 and at least one of D₁, D₂, D₃ ≠ 0, then the given system of equations is consistent and has a unique non-trivial solution.

(ii) If D ≠ 0 & D₁ = D₂ = D₃ = 0, then the given system of equations is consistent and has trivial solutions only.

(iii) If D = 0 but atleast one of D₁, D₂, D₃ is not zero then the equations are inconsistent and have no solution.

(iv) If D = D₁ = D₂ = D₃ = 0, then the given system of equations may have infinite or no solution.

Note: In case $\begin{array}{l}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{1}x+b_{1}y+c_{1}z=d_2\\ a_{1}x+b_{1}y+c_{1}z=d_3\end{array}\}$

(At Least two of d₁, d₂ & d₃ are not equal) 

D = D₁ = D₂ = D₃ = 0. But these three equations represent three parallel planes. Hence the system is inconsistent.

9.0Determinant of Matrix Solved Example

Example 1: If $\left|\begin{array}{ccc}x+3& 1& -2\\ 3& -2& 1\\ -x& -3& 3\end{array}\right|=0$ , find .

Solution:$\left|\begin{array}{ccc}x+3& 1& -2\\ 3& -2& 1\\ -x& -3& 3\end{array}\right|$=0

⇒ (x+3)[-6-(-3)]-1[9+x]-2[-9-2 x] 

⇒ (x+3)(-6+3)-(9+x)-2(-9-2 x)=0 

⇒ -3(x+3)-9-x+18+4 x=0 

⇒ -3 x-9-9-x+18+4 x=0 

⇒ -4 x+4 x-18+18=0 

∴  x $\in$ R .

Example 2: If in the determinant $\left|\begin{array}{lll}x& 3& 3\\ 3& 3& x\\ 2& 3& 3\end{array}\right|,C_{11}=C_{22}$, where C_{i j} is cofactor of element a_{i j} then x = 

(A) 2 (B) –2 (C) $\frac{5}{2}$ (D) $-\frac{9}{8}$

Ans. (C)

Solution:

$C_{11}=C_{22}\Rightarrow (-1{)}^{1+1}\left|\begin{array}{ll}3& x\\ 3& 3\end{array}\right|=(-1{)}^{2+2}\left|\begin{array}{ll}x& 3\\ 2& 3\end{array}\right|$

⇒ 9-3 x=3 x-6 $\Rightarrow$ -6 x=-15 $\Rightarrow$ x=$\frac{5}{2}$

Example 3: Find the value of the determinant $\left|\begin{array}{lll}{a}^2& ab& ac\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right|$ 

Solution:

D=$\left|\begin{array}{lll}{a}^2& ab& ac\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right|=a\left|\begin{array}{ccc}a& b& c\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right|$ = $abc\left|\begin{array}{lll}a& b& c\\ a& b& c\\ a& b& c\end{array}\right|$=0

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

(A) 2 (B) 1 (C) –1 (D) 0

Ans. (B)

Solution:

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

⇒  $\left|\begin{array}{ccc}9+k& 16& 3\\ 7& 9& 1\\ 9& 11& 1\end{array}\right|=0\left(R_3\to R_3-R_2;R_2\to R_2-R_1\right)$

⇒ k-1=0 $\Rightarrow$k=1

Example 5: Prove that $\left|\begin{array}{ccc}23& 66& 11\\ 36& 55& 26\\ 63& 143& 37\end{array}\right|$=0

Solution:

$\left|\begin{array}{ccc}23& 66& 11\\ 36& 55& 26\\ 63& 143& 37\end{array}\right|$

Taking 11 common from C₂

=$11\left|\begin{array}{ccc}23& 6& 11\\ 36& 5& 26\\ 63& 13& 37\end{array}\right|C_1\to C_1-\left(2C_2+C_3\right)$

= $11\left|\begin{array}{ccc}0& 6& 11\\ 0& 5& 26\\ 0& 13& 37\end{array}\right|$=0

Example 6: If α, β and γ are the roots of the equation x^3+p x+q=0 then the value of the determinant $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$  is-

(A) p (B) q (C) p² – 2q (D) 0

Ans. (D)

Solution:

α, β , γ are roots of x³ + px + q = 0  

∴ $\alpha +\beta +\gamma$ =0 

⇒ on $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$ , Applying $C_1\to C_1+C_2+C_3$ 

$\Rightarrow (\alpha +\beta +\gamma )\left|\begin{array}{ccc}1& \beta & \gamma \\ 1& \gamma & \alpha \\ 1& \alpha & \beta \end{array}\right|$

=0

Example 7: 

Using factor property of determinants prove that $\left|\begin{array}{lll}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|=(x-y)(y-z)(z-x)$

Solution:

On putting x=y, y=z, z=x  we are getting value of determinant is 0 so by factor theorems (x–y), (y–z) and (z–z) are factors. So, by factor theorem

$\Rightarrow \left|\begin{array}{lll}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|=\lambda (x-y)(y-z)(z-x)$

For λ put x = 0, y = 1, z = 2

We get λ = 1 

So R.H.S. = (x-y)(y-z)(z-x) 

Example 8: 

Let $\alpha \&\beta$ be the roots of equation a x^2+b x+c=0 $\text{ and }{S}_n={\alpha }^n+{\beta }^n\text{ for }n\ge 1$ . Evaluate the value of the determinant $\left|\begin{array}{ccc}3& 1+S_1& 1+S_2\\ 1+S_1& 1+S_2& 1+S_3\\ 1+S_2& 1+S_3& 1+S_4\end{array}\right|$ .

Solution:

D=

$\left|\begin{array}{ccc}3& 1+S_1& 1+S_2\\ 1+S_1& 1+S_2& 1+S_3\\ 1+S_2& 1+S_3& 1+S_4\end{array}\right|=\left|\begin{array}{ccc}1+1+1& 1+\mathrm{a}+\mathrm{b}& 1+{\mathrm{a}}^2+{\mathrm{b}}^2\\ 1+\mathrm{a}+\mathrm{b}& 1+{\mathrm{a}}^2+{\mathrm{b}}^2& 1+{\mathrm{a}}^3+{\mathrm{b}}^3\\ 1+{\mathrm{a}}^2+{\mathrm{b}}^2& 1+{\mathrm{a}}^3+{\mathrm{b}}^3& 1+{\mathrm{a}}^4+{\mathrm{b}}^4\end{array}\right|$

=$\left|\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& b& b^2\end{array}\right|\times \left|\begin{array}{ccc}1& 1& 1\\ 1& a& b\\ 1& a^2& b^2\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& b& b^2\end{array}\right|=[(1-a)(1-b)(a-b){]}^2$

$\Rightarrow D=(\alpha -\beta {)}^2(\alpha +\beta -\alpha \beta -1{)}^2$

∵ are roots of the equation

⇒ $\mathrm{a}+\mathrm{b}=\frac{-b}{a}\mathrm{ab}=\frac{c}{a}\Rightarrow |\mathrm{a}-\mathrm{b}|=\frac{\sqrt{b^2-4ac}}{|a|}$

⇒ D=$\frac{\frac{\left(b^2-4ac\right)}{a^2}{\left(\frac{a+b+c}{a}\right)}^2}{a}=\frac{\left(b^2-4ac\right)(a+b+c{)}^2}{a^4}$

Example 9: If the system of equations $x+\lambda y+1=0,\lambda x+y+1=0\&x+y+\lambda =0$  is consistent, then find the value of $\lambda$ .

Solution:

For consistency of the given system of equations

D=$\left|\begin{array}{lll}1& \lambda & 1\\ \lambda & 1& 1\\ 1& 1& \lambda \end{array}\right|$=0

⇒ 3λ = 1 + 1 + λ³ or λ³ – 3λ + 2 = 0

⇒ (λ – 1)² (λ + 2) = 0

⇒ λ = 1 or λ = –2

10.0Determinant of Matrix Practice Question

1. $\text{ Let }A=\left[\begin{array}{lll}{\cos }^{-1}x& {\cos }^{-1}y& {\cos }^{-1}z\\ {\cos }^{-1}y& {\cos }^{-1}z& {\cos }^{-1}x\\ {\cos }^{-1}z& {\cos }^{-1}x& {\cos }^{-1}y\end{array}\right]$ 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 x^3+p x+q=0, then the value of the determinant $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$=

(A)p (B) q (C) p² – 2q (D) none

3. If a, b & c  are non-zero real numbers^, then D=$\left|\begin{array}{ccc}{b}^2{c}^2& bc& b+c\\ c^2{a}^2& ca& c+a\\ a^2{b}^2& ab& a+b\end{array}\right|$= 

(A)abc (B) a²b²c² (C) bc + ca + ab (D) zero

4. The determinant  $\left|\begin{array}{lll}{b}_1+c_1& c_1+a_1& a_1+b_1\\ b_2+c_2& c_2+a_2& a_2+b_2\\ b_3+c_3& c_3+a_3& a_3+b_3\end{array}\right|$ =

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

(B) $2\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

   (C) ${}_3\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

  (D) none of these

5. 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

11.0Sample Questions on Determinant of a Matrix

Q. How do you calculate the determinant of a 2 x 2 matrix?

Ans: For a 2×2 matrix A given by:

A=$\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)$

The determinant is calculated as:

$\det (A)$ =a d-b c

Q. How do you calculate the determinant of a 3 x 3 matrix?

Ans: For a 3 × 3 matrix A given by:

A=$\left(\begin{array}{lll}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$

The determinant is calculated using the cofactor expansion along the first row:

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

Q. How do you find the determinant of a triangular matrix?

Ans: In both upper and lower triangular matrices, the determinant equals the product of their diagonal elements. For an n × n triangular matrix T with diagonal elements $t_{11},t_{22},\dots ,t_{nn}$ :

$\det (T)=t_{11}\cdot t_{22}\cdot \dots \cdot t_n$

Q. What are some properties of the determinant?

Ans: 

• Multiplicative Property: $\det (AB)=\det (A)\cdot \det (B)$
• Transpose Property: $\det \left(A^T\right)=\det (A)$
• Inverse Property: If A is invertible, $\det \left(A^{-1}\right)=\frac{1}{\det (A)}$

Row Operations: Swapping two rows of a matrix multiplies the determinant by –1, scaling a row by a factor k multiplies the determinant by k, and adding a multiple of one row to another does not change the determinant.

Q. How does the determinant relate to eigenvalues?

Ans: For a square matrix  A, the determinant is the product of its eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n:\det (A)={\lambda }_1\cdot {\lambda }_2\cdot \dots \cdot {\lambda }_n$

Q. How does one use the determinant to solve a system of linear equations?

Ans: Cramer's Rule uses the determinant to solve a system of linear equations AX = B. If A is an n × n matrix with $\det (A)\ne 0$ , the solution for the variables x_i can be found using:

$x_i=\frac{\det \left(A_i\right)}{\det (A)}$, where A_i is the matrix obtained by substituting the i^th column of A with the column vector B.