Determinant of a Matrix Formula

Determinant of a matrix is a fundamental concept in linear algebra that plays a crucial role in solving systems of linear equations, analyzing the properties of matrices, and understanding transformations in geometry. It is a scalar value that provides important properties about the matrix, such as whether it is invertible and how it scales volumes.

In this article, we will learn about the Determinant of a Matrix its definition, Formula and related concepts in detail.

What is Determinant of a Matrix?

Determinant of a matrix is a unique number associated with that square matrix. The determinant of a matrix can be calculated for only a square matrix. If A =[a_ij] is a square matrix of order n, then A's determinant is represented by det A or |A|. The general representation of the determinant of matrix A is, 

det A or |A| or \begin{vmatrix} a_{11} & a_{12} & ... & a_{1j} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2j} & ... & a_{2n}\\ .& . & ... &. & ... & .\\ .& . & ... &. & ... & .\\ a_{i1} & a_{i2} & ... & a_{ij} & ... & a_{in}\\ .& . & ... &. & ... & .\\ a_{n1} & a_{n2} & ... & a_{nj} & ... & a_{nn}\\ \end{vmatrix}

Determinant of a Matrix Formula

• Determinant of Matrix 1x1 

If A = [a_ij] is a square matrix of order 1, then the determinant of A is defined as,

|A| = a₁₁  

• Determinant of Matrix 2x2

If A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}  is a square matrix of order 2 then the determinant is defined as,

|A| = a₁₁a₂₂ - a₁₂a₂₁

• Determinant of Matrix 3x3

If A = \begin{bmatrix} a_{11} & a_{12} &a_{13}\\ a_{21} & a_{22} &a_{23}\\a_{31} & a_{32} &a_{33}\end{bmatrix}  is a square matrix of order 2 then the determinant is defined as,

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

|A| = (-1)¹⁺¹ a₁₁ \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}  +  (-1)¹⁺² a₁₂  \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}   + (-1)¹⁺³ a₁₃ \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

|A| = a₁₁  \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a₁₂   \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}  +a₁₃ \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

|A| = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₃₃a₂₁ - a₂₃a₃₁) + a₁₃(a₃₂a₂₁ - a₂₂a₃₁) ⇢ (1)

|A| = a₁₁a₂₂a₃₃ + a₁₂a₂₃a₃₁ + a₁₃a₃₂a₂₁ - a₁₁a₂₃a₃₂ - a₂₂a₁₃a₃₁ - a₁₂a₂₁a₃₃

Thus, determinant of a square matrix of order 3 is the sum of the product of elements a_ij in i^th row with (-1)^i+j  times the determinant of a 2 x 2 sub-matrix obtained by leaving the i^th row and j^th column passing through the element. The expansion is done through the elements of i^th row. Then, it is known as the expansion along the i^th row. The above expansion (1) of |A| is known as the expansion along the first row.

Note:

• Only square matrices have determinants. Matrices which are not square do not have determinants.
• Determinant of order three can be expanded along any row and column.
• If a row or column of a determinant is all zeros, then the determinant is zero.

Minor of a Matrix

Consider a square matrix A =[a_ij] of order n. The minor M_ij of a_ij in A determines the square sub-matrix of order (n - 1) obtained by leaving the ith row and jth column of A.

Cofactor of a Matrix

Consider a square matrix A = [aij] of order n. The cofactor C_ij of a_ij in A is equal to (-1)i + j times the minor M_ij of a_ij.

C_ij = Cofactor of a_ij in A = (-1)^i+j M_ij    

Where M_ij is the minor of a_ij in A.

C_ij = \begin{cases} \hspace{0.2cm}M_{ij}, & \text{if $n$ is even}\\ -M_{ij}, & \text{if $n$ is odd} \end{cases}

Singular Matrix of a Matrix

A square matrix is said to be singular matrix if its determinant is zero and if the determinanat of the matrix is not equal to zero then it is called non-singular.

Properties of Determinant

Let A be a square matrix of order n,

• Sum of product of elements of any row(column) with their cofactors is equal to |A|.   \sum_{j=1}^n a_{ij}C_{ij} = |A|  and \sum_{i=1}^n a_{ij}C_{ij} = |A|
• Sum of product of elements of any row(column) with the cofactors of the corresponding elements of some other row(column) is zero.  \sum_{j=1}^n a_{ij}C_{kj} = 0  and  \sum_{i=1}^n a_{ij}C_{ik} = 0 
• Determinant remains unchanged if its rows and columns are interchanged(transpose). |A| =|A^T| 
• If two rows (columns) are interchanged, the determinant changes by a minus sign. |B| = -|A| (B is a matrix obtained by interchanging two rows (columns) of A).
• If two rows or columns of A are the same, then its determinant is zero. |A| = 0
• If B is the matrix obtained by multiplying scalar value k in each row element or column of matrix A then |B| = k|A|.
• If each row element (or column) is zero, then the determinant is zero.
• If A is a diagonal matrix, then |A| = a₁₁ × a₂₂ × .... a_nn.
• If A and B are square matrices of the same order, then |AB| = |A||B|.
• If each element of a row (column) can be expressed as the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.
• If B is a matrix obtained by adding to a row (column) of A a scalar multiple of another row(column) of A, then |B| = |A|.

Examples problems based on Determinant of a Matrix Formula

Example 1: Evaluate:

\begin{vmatrix} -2 & 6 \\ 4 & 3 \end{vmatrix}   

\begin{vmatrix} sin\theta & cos\theta \\ -cos\theta & sin\theta \end{vmatrix} 

Solution:

Given:

\begin{vmatrix} -2 & 6 \\ 4 & 3 \end{vmatrix} 

= (-2) × 3 - 6 × 4 = -30

\begin{vmatrix} sin\theta & cos\theta \\ -cos\theta & sin\theta \end{vmatrix}

= sin²θ - (-cos²θ) = sin²θ + cos²θ = 1

Example 2: Evaluate:

(i) D = \begin{vmatrix} -3 & 2 &4\\ 6& 9&5\\1 & 2 &3\end{vmatrix} expand along first row.

(ii) D= \begin{vmatrix} -1 & 2 &0\\ 3&5&4\\1 & -2 &1\end{vmatrix} expand along second column.

Solution:

(i)

D= \begin{vmatrix} -3 & 2 &4\\ 6& 9&5\\1 & 2 &3\end{vmatrix}

= (-1)^{1 + 1}(-3)\begin{vmatrix} 9 & 5 \\ 2& 3 \end{vmatrix} +  (-1)¹⁺²(2)\begin{vmatrix} 6 & 5 \\ 1 & 3 \end{vmatrix} + (-1)¹⁺³(4) \begin{vmatrix} 6 & 9 \\ 1 & 2 \end{vmatrix}

= -3\begin{vmatrix} 9 & 5 \\ 2& 3 \end{vmatrix} - 2\begin{vmatrix} 6 & 5 \\ 1 & 3 \end{vmatrix}    + 4\begin{vmatrix} 6 & 9 \\ 1 & 2 \end{vmatrix}

= -3[9 × 3 - 5 × 10 ] - 2[6 × 3 - 5 × 1] + 4[6 × 2 - 9 × 1]

= -3[27 -10] - 2[18 -5] + 4[12 -9]

= -3 × 17 - 2 × 13 + 4 × 3

= -51 -26 +12 = -65

(ii)

D =\begin{vmatrix} -1 & 2 &0\\ 3&5&4\\1 & -2 &1\end{vmatrix}

= (-1)^{1 + 2} (2) \begin{vmatrix} 3 & 4 \\ 1& 1 \end{vmatrix} +  (-1)²⁺²(5)\begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix}    + (-1)³⁺²(-2)\begin{vmatrix} -1 & 2 \\ 3 & 5 \end{vmatrix}

= -2\begin{vmatrix} 3 & 4 \\ 1& 1 \end{vmatrix}  + 5\begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix}    + 2\begin{vmatrix} -1 & 2 \\ 3 & 5 \end{vmatrix}

= -2[3 × 1 - 4 × 1] + 5[1 × 1 - 1 × 0] + 2[(-1) × 5 - 2 × 3]

= -2[3 - 4] + 5[1 - 0] + 2[-5 - 6]

= (-2) × (-1) + 5 × 1 + 2 × (-11)

= (-2) × (-1) + 5 × 1 + 2 × (-11)

= 2 + 5 - 22 = -15

Example 3: Evaluate, D = \begin{vmatrix}1 & 0 &2 &0\\ 3& -1 & 8 & 6\\0 & -4 &-2& 3\\2&6&7&1 \\\end{vmatrix}  expand along first row.

Solution:

D = \begin{vmatrix}1 & 0 &2 &0\\ 3& -1 & 8 & 6\\0 & -4 &-2& 3\\2&6&7&1 \\\end{vmatrix}

D = (-1)¹⁺¹(1)\begin{vmatrix}\\ -1 & 8 & 6\\ -4 &-2& 3\\6&7&1 \\\end{vmatrix}   + (-1)¹⁺²(0) \begin{vmatrix}\\ 3&  8 & 6\\0 & -2& 3\\2&7&1 \\\end{vmatrix}   + (-1)¹⁺³(2) \begin{vmatrix}\\ 3& -1 &  6\\0 & -4 & 3\\2&6&1 \\\end{vmatrix}  + (-1)¹⁺⁴(0)\begin{vmatrix}\\ 3& -1 & 8 \\0 & -4 &-2\\2&6&7 \\\end{vmatrix}

D = 1[(-1)¹⁺¹(-1) \begin{vmatrix}\\-2& 3\\7&1 \\\end{vmatrix}  + (-1)¹⁺²(8)\begin{vmatrix}\\-4 & 3\\6&1 \\\end{vmatrix}  +(-1)¹⁺³(6) \begin{vmatrix}\\ -4 &-2\\6&7 \\\end{vmatrix}  ] - 0 +2[(-1)¹⁺¹(3)\begin{vmatrix}\\-4 & 3\\6&1 \\\end{vmatrix}   + (-1)¹⁺²(-1) \begin{vmatrix}\\0 & 3\\2&1 \\\end{vmatrix}   +(-1)¹⁺³(6)\begin{vmatrix}\\0 & -4\\2&6 \\\end{vmatrix}  ] - 0

D = 1[(-1){(-2} × 1 - 3 × 7} - 8{(-4) × 1 - 6 × 3} + 6{(-4) × 7 - 6 × (-2)}] + 2[3{(-4} × 1 - 6 × 3} + 1{0 × 1 - 3 × 2} + 6{0 × 6 -(-4) × 2}]

D = [-1 × (-2 - 21) - 8 × (-4 - 18) + 6(-28 + 12)] + 2[3 × (-4 - 18) + (0 - 6) + 6 × (0 + 8)]

D = [-1 × (-23) - 8 × (-22) + 6 × (-16)] + 2[3 × (-22) -6 + 48]

D = [23 + 176 - 96] + 2[-66 - 6 + 48]

D = 103 + 2 × (-24) = 103 - 48

D = 55 

Example 4: Find the value of x for which matrix A = \begin{bmatrix} 1 & -2 &3\\ 1&2 & 1\\x& 2& -3 \end{bmatrix}  is singular. 

Solution: 

Matrix A is singular, if |A| = 0

\begin{vmatrix} 1 & -2 &3\\ 1&2 & 1\\x& 2& -3 \end{vmatrix}  = 0 

⇒ 1\begin{vmatrix} 2 & 1\\ 2&-3  \end{vmatrix}  + 2\begin{vmatrix} 1 & 1\\ x&-3  \end{vmatrix}  + 3\begin{vmatrix} 1 & 2\\ x&2  \end{vmatrix}  = 0

⇒ (-6 - 2) + 2(-3 - x) + 3(2 - 2x) = 0

⇒ -8 - 6 - 2x + 6 - 6x = 0

⇒ -8x - 8 = 0

⇒ x = -1     

Example 5: Show that \begin{vmatrix} 1 & a &b+c\\ 1&b & a+c\\1& c& a+b \end{vmatrix} =0.

Solution: 

D = \begin{vmatrix} 1 & a &b+c\\ 1&b & a+c\\1& c& a+b \end{vmatrix}     

Apply C₂ ⇢ C₂ + C₃

D = \begin{vmatrix} 1 & a+b+c &b+c\\ 1&b+a+c & a+c\\1& c+a+b& a+b \end{vmatrix} 

D = (a+ b+ c)\begin{vmatrix} 1 & 1 &b+c\\ 1&1& a+c\\1& 1& a+b \end{vmatrix} 

D = (a+ b+ c).0 = 0 [Since, C₁ and C₂ are identical so determinant = 0 (Property of determinant)]

Example 6: Evaluate : \begin{vmatrix} 1 & 2 &3\\ 2&4 & 6\\5&9&7\end{vmatrix} 

Solution: 

D = \begin{vmatrix} 1 & 2 &3\\ 2&4& 6\\5&9&7\end{vmatrix} 

R₂ ⇢ R₂ - 2R₁

D = \begin{vmatrix} 1 & 2 &3\\ 0&0&0\\5&9&7\end{vmatrix} 

[Since, all the elements of R₂ are 0 so, D = 0]

D = 0

Example 7: Prove that :  \begin{vmatrix}a+b & a & a\\ 5a+4b& 4a & 2a \\10a+8b & 8a &3a \\\end{vmatrix}   = a³ 

Solution: 

Let  D = \begin{vmatrix}a+b & a & a\\ 5a+4b& 4a & 2a \\10a+8b & 8a &3a \\\end{vmatrix}

Since, C₂ contains sum of two elements it can be represented as sum of two determinants.

D = \begin{vmatrix}a & a & a\\ 5a& 4a & 2a \\10a & 8a &3a \\\end{vmatrix} + \begin{vmatrix}b & a & a\\ 4b& 4a & 2a \\8b & 8a &3a \\\end{vmatrix}   

Taking a and b common from rows and columns.           

D = a³ \begin{vmatrix}1 & 1 & 1\\ 5& 4 & 2 \\10 & 8 &3 \\\end{vmatrix}    +ba²  \begin{vmatrix}1& 1 & 1\\ 4& 4 & 2 \\8 & 8 &3\\\end{vmatrix}

Since C₁ and C₂ in the second determinant are the same, its determinant is zero.

D = a³  \begin{vmatrix}1 & 1 & 1\\ 5& 4 & 2 \\10 & 8 &3 \\\end{vmatrix}  + ba² × 0   

Applying C₁ ⇢ C₁ - C₃, C₂ ⇢ C₂ - C₃ 

D = a³  \begin{vmatrix}0 & 0 & 1\\ 3& 2 & 2 \\7 & 5 &3 \\\end{vmatrix}

D = a³× 1 × \begin{vmatrix} 3& 2  \\7 & 5  \\\end{vmatrix}

D = a³ (15 - 14) = a³

Practice Problems on Determinant of a Matrix Formula

1. Calculate the determinant of the following 2x2 matrix:A=\begin {bmatrix}3&5\\7&2 \end {bmatrix}
2. Find the determinant of the 3x3 matrix:A=\begin {bmatrix}1&2&3\\4&5&6\\7&8&9 \end {bmatrix}
3. Determine the value of the determinant for the following 2x2 matrix: A=\begin {bmatrix}-4&6\\2&-3 \end {bmatrix}
4. Find the determinant of the 3x3 matrix: A=\begin {bmatrix}2&0&1\\3&-2&4\\-1&5&7 \end {bmatrix}
5. Find the determinant of the matrix: A=\begin {bmatrix}0&1\\1&0\end {bmatrix}
6. Given the matrix below, calculate its determinant: A=\begin {bmatrix}2&3&1\\0&-1&4\\5&2&3\end {bmatrix}
7. If B=\begin {bmatrix}5&7\\2&3 \end {bmatrix}, determine the determinant of 3B.
8. Find the determinant of the 3x3 identity matrix: I_{3}=\begin {bmatrix}1&0&0\\0&1&0\\0&0&1 \end {bmatrix}
9. Given the matrix A= \begin {bmatrix}1&2\\3&6 \end{bmatrix}, find the determinant and interpret its meaning.
10. Calculate the determinant of the following 3x3 matrix: A=\begin {bmatrix}2&0&-1\\3&1&4\\0&-2&5 \end {bmatrix}

Answers: 
1) -29 ; 
2) 0 ; 
3) 0 ; 
4) -55 ; 
5) -1 ; 
6 )43 ;
7) 9 ; 
8) 1 ; 
9) 0, which means the matrix is singular (non-invertible) and its rows are linearly dependent. ; 
10) 32

Conclusion

The determinant of a matrix is a fundamental concept in linear algebra with broad applications in various fields, including physics, engineering, and computer science. Understanding how to calculate and interpret the determinant is crucial for analyzing the properties of matrices, such as invertibility and the scaling of geometric objects. Whether working with small matrices using direct formulas or larger ones using co-factor expansion, mastering the determinant will be a powerful tool for solving complex problems in mathematics and beyond.

Also Read,

• Determinant of a Matrix
• Determinant of 3×3 Matrix
• How to find the Determinant of a Matrix?
• Determinant of a Square Matrix
• Practice Problems on Determinant using Elementary Operations