Determinant of a Matrix is defined as the function that gives the unique output (real number) for every input value of the matrix. Determinant of the matrix is considered the scaling factor that is used for the transformation of a matrix. It is useful for finding the solution of a system of linear equations, the inverse of the square matrix, and others. The determinant of only square matrices exists.

Table of Content

- Determinant of Matrix Calculator
- Definition of Determinant of Matrix
- Determinant of a 1×1 Matrix
- Determinant of 2×2 Matrix
- Determinant of a 3×3 Matrix
- Determinant of 4×4 Matrix
- Determinant of Identity Matrix
- Determinant of Symmetric Matrix
- Determinant of Skew-Symmetric Matrix
- Determinant of Inverse Matrix
- Determinant of Orthogonal Matrix
- Physical Significance of Determinant
- Laplace Formula for Determinant
- Properties of Determinants of Matrix

## Determinant of Matrix Calculator

A determinant of a matrix calculator is a tool used to compute the determinant of a given matrix quickly and accurately.

## Definition of Determinant of Matrix

Determinant of a Matrix is defined as the sum of products of the elements of any row or column along with their corresponding co-factors. Determinant is defined only for square matrices.

Determinant of any square matrix of order 2Ã—2, 3Ã—3, 4Ã—4, or n Ã— n, where n is the number of rows or the number of columns. (For a square matrix number of rows and columns are equal). Determinant can also be defined as the function which maps every matrix with the real numbers.Â

For any set S of all square matrices, and R the set of all numbers the function f, f: S â†’ R is defined as f (x) = y, where x âˆˆ S and y âˆˆ R, then f (x) is called the determinant of the input matrix.

### Symbol of Determinant

Let’s take any square matrix A, then the determinant of A is denoted as ** det A** (or)

**. Determinant is also denoted by the symbol**

**|A|****.**

**Î”****Minor of Element of Matrix**

**Minor of Element of Matrix**

Minor is required to find determinant for single elements (every element) of the matrix. They are the determinants for every element obtained by eliminating the rows and columns of that element. If the matrix given is:

[Tex]\begin{bmatrix}a_{11} & a_{12} &a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix}[/Tex]

Minor of a_{12 }will be the determinant:

[Tex]\begin{vmatrix}a_{21} & a_{23}\\a_{31} & a_{33}\end{vmatrix}[/Tex]

**Question:**** Find the Minor of element 5 in the determinant**Â [Tex]\begin{vmatrix}2 & 1 & 2\\4 & 5 & 0\\2 & 0 & 1\end{vmatrix}[/Tex]

**Answer:**

The minor of element 5 will be the determinant of

[Tex]\begin{vmatrix}2 & 2\\2 & 1\end{vmatrix}[/Tex]ÂCalculating the determinant, the minor is obtained as:Â

(2 Ã— 1) – (2 Ã— 2) = -2

**Cofactors of Element of Matrix**

**Cofactors of Element of Matrix**

Cofactors are related to minors by a small formula, for an element a_{ij}, the cofactor of this element is C_{ij} and the minor is M_{ij} then, the cofactor can be written as:Â

C_{ij }= (-1)^{i+j}M_{ij}

**Question:****Find the cofactor of the element placed in the first row and second column of the determinant:**

[Tex]\begin{vmatrix}2 & 1 & 2\\4 & 5 & 0\\2 & 0 & 1\end{vmatrix}[/Tex]

**Answer:Â **

In order to find out the cofactor of the first row and second column element i.e the cofactor for 1. First find out the minor for 1, which will be:Â

[Tex]\begin{vmatrix}4 & 0\\2 & 1\end{vmatrix} \\ = (4 \times 1) – (2 \times 0) \\ = 4[/Tex]

M

_{12}= 4Now, applying the formula for cofactor:Â

C

_{12 }= (-1)^{1 + 2}M_{12}C

_{12}= (-1)^{3}Ã— 4C

_{12}= -4

**Adjoint of a Matrix**

**Adjoint of a Matrix**

The Adjoint of a matrix for order n can be defined as the transpose of its cofactors. For a matrix A:

Adj. A = [C_{ij}]_{nÃ—n}^{T}

**Transpose of a Matrix**

**Transpose of a Matrix**

Transpose of a Matrix A is denoted as A^{T} or A’. It is clear that the vertical side in the matrix is known as a column and the horizontal side is known as a row, Transposing a Matrix means replacing the Rows with columns and Vice-Versa, since the Rows and Columns are changing, the Order of the Matrix also changes.

Â If a Matrix is given as A= [a

_{ij}]_{mÃ—n}, then its Transpose will become

A^{T}or A’ = [a_{ji}]_{nÃ—m}

**Question: What will be the transpose of the Matrix A =**

[Tex]\begin{bmatrix}2 & 1\\3 & 0\\6 & 9\end{bmatrix}_{2\times3}[/Tex]

** Answer:**Â

Interchanging Rows and Columns, A

^{T}=Â [Tex]\begin{bmatrix}2 & 3 & 6\\1 & 0 & 9\end{bmatrix}_{3\times2}[/Tex]

**Determinant of a 1×1 Matrix**

**Determinant of a 1×1 Matrix**

Let X = [a] be the matrix of order one, then its determinant is given by det(X) = a.Â

## Determinant of 2×2 Matrix

The determinant of any 2Ã—2 square matrix A =Â [Tex]\begin{bmatrix}a & b\\c & d\end{bmatrix}_{2\times2}Â Â [/Tex]

is calculated using the formula ** |A| = ad – bc**.Â

**Example: Find the Determinant of A =Â **

**Solution:**

Determinant of A =Â [Tex]\begin{bmatrix}3 & 2\\2 & 3\end{bmatrix}_{2\times2}Â Â [/Tex]Â is calculated as,

| A | =Â [Tex]\begin{vmatrix}3 & 2\\2 & 3\end{vmatrix}[/Tex]

| A | = 3Ã—3 – 2Ã—2Â

Â Â Â Â = 9 – 4Â

Â Â Â Â = 5

**Determinant of a 3×3 Matrix**

**Determinant of a 3×3 Matrix**

Determinant of a 3×3 Matrix is determined by expressing it in terms of 2nd-order determinants. It can be expanded either along rows(R_{1}, R_{2} or R_{3}) or column(C_{1}, C_{2} or C_{3}). Consider a Â matrix A of order 3Ã—3Â

det(A) =a_{11â€‹}(a_{22â€‹}a_{33}â€‹ –a_{23}â€‹a_{32}â€‹) –a_{12}â€‹(a_{21â€‹}a_{33â€‹ }–a_{23}â€‹a_{31}â€‹) +a_{13}â€‹(a_{21}â€‹a_{32}â€‹ –a_{22}â€‹a_{31}â€‹)

**How to Find Determinant of 3×3 Matrix**

**How to Find Determinant of 3×3 Matrix**

For calculating the Determinant of 3×3 Matrix use the following step:

Multiply the first element aStep 1:_{11}of row R_{1}with (-1)^{(1 + 1)}[(-1)sum of suffixes in a_{11}] and with the second order determinant obtained by deleting the elements of row R_{1}and C_{1}of A as a_{11}lies in R_{1}and C_{1}.Â [Tex](-1)^{1 + 1}a_{1}\begin{vmatrix} b_{2} & b_{3} \\ c_{2} & c_{3} \end{vmatrix}[/Tex]

Similarly, multiply the second element of the first rowRStep 2:_{1}, Â with the determinant obtained after deleting the first row and second column.Â [Tex](-1)^{1 + 2}a_{12}\begin{vmatrix} b_{1} & b_{3} Â \\ c_{1} & c_{3} \end{vmatrix}[/Tex]

Multiply the third element of row RStep 3:_{1}with the determinant obtained after deleting the first row and third column.Â [Tex](-1)^{1 + 3}a_{3}\begin{vmatrix} b_{1} & b_{2} \\ c_{1} & c_{2} \end{vmatrix}[/Tex]

Now the expansion of the determinant of A, that is |A| can be written as |A| = Â [Tex](-1)^{1 + 1}a_{1}\begin{vmatrix} b_{2} & b_{3} \\ c_{2} & c_{3} \end{vmatrix} + (-1)^{1 + 2}a_{12}\begin{vmatrix} b_{1} & b_{3} Â \\ c_{1} & c_{3} \end{vmatrix} + (-1)^{1 + 3}a_{3}\begin{vmatrix} b_{1} & b_{2} \\ c_{1} & c_{2} \end{vmatrix}[/Tex]Step 4:

Similarly, in this way, we can expand it along any row and any column.Â

**Example:**** Evaluate the determinant det(A) =Â **[Tex]\begin{vmatrix} 1 & 3 & 0 \\ 4 & 1 & 0 \\ 2 & 0 & 1 \end{vmatrix}[/Tex]

** Solution:**Â

We see that the third column has most number of zeros, so it will be easier to expand along that column.Â

det(A) =Â [Tex](-1)^{1 + 3}0\begin{vmatrix}4 & 1 \\ 2 & 0 \end{vmatrix} + (-1)^{2 + 3}0\begin{vmatrix}1 & 3 \\ 2 & 0 \end{vmatrix} Â + (-1)^{1 + 3}1\begin{vmatrix}1 & 3 \\ 4 & 1 \end{vmatrix} \\ = -11[/Tex]

**Determinant of 4×4 Matrix**

**Determinant of 4×4 Matrix**

Determining the determinant of a 4×4 matrix involves more complex methods such as expansion by minors or Gaussian elimination. These techniques require breaking down the matrix into smaller submatrices and recursively finding their determinants. While there isn’t a direct formula like Sarrus’ Rule for 3×3 matrices, the process involves systematic calculations based on the properties of determinants.

### How to Find Determinant of 4×4 Matrix

By following these step-by-step instructions, you can efficiently find the determinant of a 4×4 matrix using cofactor expansion:

: Choose a row or column along which you will perform cofactor expansion. While any row or column can be chosen, it’s often easiest to select the row or column with the most zeros to simplify calculations.**Identify a Row or Column for Expansion**: For each element**Calculate Cofactors***a*_{ij}_{â€‹}of the chosen row or column, calculate its cofactor*C*â€‹. To find the cofactor, follow these steps:_{ij}- For each element
*a*â€‹, find the determinant of the 3×3 submatrix formed by eliminating the row and column containing_{ij}*a*_{ij}_{â€‹}. - Apply the alternating sign pattern: positive for elements in even positions and negative for elements in odd positions.
- Multiply the determinant of the 3×3 submatrix by the appropriate sign to obtain the cofactor
*C*_{ij}_{â€‹}.

- For each element
: Multiply each element of the chosen row or column by its corresponding cofactor.**Multiply Elements by Their Cofactors**: Sum up the products obtained in the previous step. This will give you the determinant of the 4×4 matrix.**Sum Up the Results**: If needed, repeat the process of cofactor expansion along other rows or columns until you find the determinant. However, it’s generally most efficient to choose the row or column with the most zeros initially.**Repeat for Other Rows or Columns (Optional)**: Double-check your calculations and verify the determinant obtained.**Verify the Determinant**: The final result is the determinant of the 4×4 matrix.**Result**

## Determinant of Identity Matrix

An identity matrix is a square matrix in which all the elements of the main diagonal are ones, and all other elements are zeros. For example, a 3×3 identity matrix looks like this:

[Tex]I =\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)[/Tex]

The determinant of an identity matrix of any size is always 1. This property can be understood intuitively by considering that the identity matrix represents a transformation that leaves vectors unchanged when multiplied by it. Since the determinant measures how a matrix scales the space, the determinant of an identity matrix, which doesn’t scale the space at all, is 1.

Mathematically, we can express this as:

det (I) = 1

## Determinant of Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. In other words, if * A* is a symmetric matrix, then

*=*

*A*

*A**. Symmetric matrices have several interesting properties, one of which is that their determinants remain unchanged under transpose.*

^{T}Hence, for a symmetric matrix * A*, we have:

det(A) = det(A^{T})

This property simplifies the computation of determinants for symmetric matrices since you can work with either the original matrix or its transpose, whichever is more convenient.

## Determinant of Skew-Symmetric Matrix

A skew-symmetric (or antisymmetric) matrix is a square matrix whose transpose is equal to its negative. In other words, if * A* is a skew-symmetric matrix, then

*= âˆ’*

*A*

*A**. Skew-symmetric matrices have interesting properties, one of which is that their determinants have specific values based on the order of the matrix.*

^{T}For skew-symmetric matrices of odd order, the determinant is always 0. This is because the determinant of a skew-symmetric matrix is always the square of its eigenvalues, and a non-zero square is always positive. Since the order of the matrix is odd, at least one eigenvalue must be zero, resulting in a determinant of 0.

For skew-symmetric matrices of even order, the determinant is a non-zero value, which can be calculated based on the elements of the matrix. However, determining the exact value typically involves more complex methods such as cofactor expansion or using properties of determinants.

## Determinant of Inverse Matrix

To understand the determinant of the inverse matrix, let’s first define what the inverse of a matrix is.

The inverse of a square matrix * A*, denoted as

*âˆ’1, is a matrix such that when it’s multiplied by*

*A**, the result is the identity matrix*

*A**. Mathematically, if*

*I***, then**

**Aâ‹…Aâˆ’1 = I****is the inverse of**

**Aâˆ’1***.*

*A*Now, the determinant of the inverse matrix, denoted as det(* A*âˆ’1), is related to the determinant of the original matrix

*. Specifically, it can be expressed by the formula:*

*A*

det(A^{âˆ’1}) = 1/det(A)

This formula illustrates an important relationship between the determinants of a matrix and its inverse. If the determinant of * A* is non-zero, meaning

**, then the inverse matrix exists, and its determinant is the reciprocal of the determinant of**

**det(A)â‰ 0***. Conversely, if*

*A*

**det(**

**A****, the matrix**

**)=0***is said to be singular, and it does not have an inverse.*

*A*Here are some key points about the determinant of the inverse matrix:

: For non-singular matrices (those with non-zero determinants), their inverses exist, and the determinant of the inverse is the reciprocal of the determinant of the original matrix.**Non-Singular Matrices**: Singular matrices (those with zero determinants) do not have inverses. Attempting to find the inverse of a singular matrix results in an undefined or non-existent inverse.**Singular Matrices**: The determinant of a matrix measures how it scales the space. Similarly, the determinant of the inverse matrix measures the scaling effect of the inverse transformation. If the original transformation expands the space, its inverse contraction will be inversely proportional, and vice versa.**Geometric Interpretation**

## Determinant of Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning that the dot product of any two distinct rows or columns equals zero, and the dot product of each row or column with itself equals one. Mathematically, if * A* is an orthogonal matrix, then

*A**â‹…*

^{T}*=*

*A**, where*

*I*

*A**denotes the transpose of*

^{T}*and*

*A**represents the identity matrix.*

*I*The determinant of an orthogonal matrix has a special property:

det (A) = Â±1

The determinant of an orthogonal matrix is either +1+1 or âˆ’1âˆ’1. This property arises from the fact that the determinant represents the scaling factor of the matrix transformation. Since orthogonal transformations preserve lengths, the determinant must be either positive (for preserving orientation) or negative (for reversing orientation).

The determinant of an orthogonal matrix being +1+1 implies that the transformation preserves orientation, while a determinant of âˆ’1âˆ’1 indicates a transformation that reverses orientation.

**Physical Significance of Determinant**

**Physical Significance of Determinant**

Consider a 2D matrix, each column of this matrix can be considered as a vector on the x-y plane. Â So, the determinant between two vectors on a 2d plane gives us the area enclosed between them. If we extend this concept, in 3D the determinant will give us the volume enclosed between two vectors.Â

## Laplace Formula for Determinant

Laplaceâ€™s formula, is used to expressed the determinant of a matrix in terms of the minors of the matrix.

If Â A_{nÃ—n} is the given square matrix and C_{ij }is the cofactor of A_{ij }the solving for any row ** i** or column

**j**det (A) =Â [Tex]\sum_{i =1}^{n}A_{ij}C_{ij}[/Tex]

## Properties of Determinants of Matrix

Various Properties of the Determinants of the square matrix are discussed below:

: Value of the determinant remains unchanged even after rows and columns are interchanged. That determinant of a matrix and its transpose remains the same.**Reflection Property**If any two rows or columns of a determinant are interchanged, then the sign of the determinant changes.**Switching Property:**: If each element in a row or column of a matrix**Scalar Multiplication Property**is multiplied by a scalar*A*, then the determinant of the resulting matrix is*k*times the determinant of*k*. Mathematically, if*A*is the matrix obtained by multiplying each element of a row or column of*B*by*A*, then*k***det(****B****) =****k****â‹…det(****A**.**)**: The determinant of the sum of two matrices**Additivity Property**and*A*is equal to the sum of their determinants. Symbolically,*B***det(****A****+****B****) = det(****A****) + det(****B**. However, this property applies only if the matrices have the same dimensions.**)**: The determinant of the product of two matrices**Multiplicative Property**and*A*is equal to the product of their determinants. Symbolically,*B***det(****AB****) = det(****A****)â‹…det(****B**. However, this property holds true only for square matrices.**)**: The determinant of a matrix**Determinant of Transpose**is equal to the determinant of its transpose*A**A*. Mathematically,^{T}**det(****A****) = det(****A**^{T}.**)**

** Example:**Â [Tex]\begin{vmatrix} 3 & 3 & 0 \\ 2 & 1 & 1 \\ 5 & 0 & 1 \end{vmatrix}[/Tex]

**Solution:**

det. A = [3Ã—{(1Ã—1)-(0Ã—1)}]-[3Ã—{(2Ã—1)-(5Ã—1)}]+[0Ã—{(2Ã—0)-(5Ã—1)}]

= {3Ã—(1-0)}-{3Ã—(2-5)+0

= [3-{3(-3)}+0]

= (3+9)

=12

Now, Interchanging Row 1 with Row 2, determinant will be:

[Tex]\begin{vmatrix} 2 & 1 & 1 \\ 3 & 3 & 0 \\ 5 & 0 & 1 \end{vmatrix}[/Tex]

det. A = [2Ã—{(3Ã—1)-(0Ã—0)}]-[1Ã—{(3Ã—1)-(5Ã—0)}]+[1Ã—{(3Ã—0)-(5Ã—3)}]

= (6-3-15)

= -12

If any two rows or any two columns of a determinant are identical, then the value of the determinant becomes zero.**Repetition Property/Proportionality Property:**If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k**Scalar Multiple Property:**

Â [Tex]\begin{vmatrix} ka & kb \\ c & d \end{vmatrix} = k\begin{vmatrix} a & b \\ c & d \end{vmatrix}Â [/Tex]

: If some or all elements of a row or column can be expressed as the sum of two or more terms, then the determinant can also be expressed as the sum of two or more determinants.**Sum Property**

Â [Tex]\begin{vmatrix} a_{1} + \lambda_{1} & a_{2} + \lambda_{2} & a_{3} + \lambda_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix} = \begin{vmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix} + \begin{vmatrix} \lambda_{1} & \lambda_{2} & \lambda_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix}Â [/Tex]

**Solved Questions on Determinant of Matrix**

**Solved Questions on Determinant of Matrix**

**Question 1:**** If x, y, and z are different. and A =**Â [Tex]\begin{vmatrix} x & x^{2} & 1 + x^{3} \\ y & y^{2} & 1 + y^{3} \\ z & z^{2} & 1 + z^{3} \end{vmatrix} = 0Â Â Â [/Tex],

**then show that 1 + xyz = 0.Â****Solution:Â **

UsingÂ Sum Property

[Tex]\begin{vmatrix} x & x^{2} & 1 + x^{3} \\ y & y^{2} & 1 + y^{3} \\ z & z^{2} & 1 + z^{3} \end{vmatrix} = \begin{vmatrix} x & x^{2} & 1 \\ y & y^{2} & 1\\ z & z^{2} & 1 \end{vmatrix} Â + \begin{vmatrix} x & x^{2} & x^{3} \\ y & y^{2} & y^{3} \\ z & z^{2} & z^{3} \end{vmatrix} \text{} \\ = (-1)^{2}\begin{vmatrix} 1 & x & x^{2} \\ 1 & y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} + xyz\begin{vmatrix} 1 & x & x^{2} \\ 1 & y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \\ = (1 + xyz) \begin{vmatrix} 1 & x & x^{2} \\ 1 & y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} = 0Â Â Â [/Tex]ÂÂOn solving this determinant and expanding it,Â

A = (1 + xyz)(y- x)(z-y)(z-x)Â

Since it’s given in the question, that all x, y and z have different values and A =0. So the only term that can be zero is 1 + xyz.Â

Hence, 1 + xyz = 0

** Question 2: EvaluateÂ **[Tex]\begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix}Â Â Â [/Tex]

**.Â****Solution:Â **

Using Scalar Multiple Property and Repetition Property

[Tex]\begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} = \begin{vmatrix} 17(6) & 6(3) & 6(6) \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} \\ = 6\begin{vmatrix} 17 & 3 & 6 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \\ \end{vmatrix} = 0 Â \text{}[/Tex]Â

** Question 3: Evaluate the determinantÂ **Â [Tex]A = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 5 \\ 2 & 3 & 1 \end{vmatrix} \\Â [/Tex]

**Solution:Â **

Using Proportionality PropertyTwo of the rows of the matrix are identical.Â

So,Â [Tex]A = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 5 \\ 2 & 3 & 1 \end{vmatrix} \\ = 0 \text{}[/Tex]

Related Articles | |
---|---|

## FAQs on Determinant of Matrix

**What is meant by the determinant formula?**

**What is meant by the determinant formula?**

For any 3Ã—3 matrix A =Â [Tex]\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \\Â Â [/Tex]Â the shortcut formula for computing its determinant is:Â

|A| = a (ei âˆ’ fh) âˆ’ b (di âˆ’ fg) + c (dh âˆ’ eg)

**Can determinant of any matrix be negative?**

**Can determinant of any matrix be negative?**

Yes, the determinant of any matrix can be negative.

**Can determinant of any matrix ever be equal to 0?**

**Can determinant of any matrix ever be equal to 0?**

Yes, the determinant of any matrix can be zero if any one row or column of the matrix has all the zero values. It can also be zero if any two rows or columns of the matrix are equal.

**How to find the determinant of matrix?**

**How to find the determinant of matrix?**

The determinant of any matrix can be found by using the following steps:

Select any row or column of our choice.Step 1:

Calculate the cofactors of all the elements of the selected row or columnStep 2:

The product of the elements of the row or column by their corresponding cofactors is found. The calculated product is added with alternate negative sign.Step 3: