Determinant of a Matrix

Determinant dates back to 1841 when Authur Cayley developed this system for solving linear equations quickly using two vertical line notations. For a given square matrix, the determinant of that matrix can compute its scalar value. The square matrix can be of any order such as 2x2 matrix, 3x3 matrix, or other nxn matrices. The important point to note here is the number of columns being equal as the number of rows. A determinant is represented with two vertical lines that consist of rows and columns. It is also represented as |A| or get A or det (A). 


The process of calculating a determinant is also discussed in this article. Follow the below-mentioned steps for calculating the value of a determinant.


Calculating the Value of a Determinant

|A| or det(A) is given below


\[\begin{vmatrix}a &b \\ c& d \end{vmatrix}\]


Step 1: 

Solving for determinants involves the multiplication of rows with columns. 

Step 2:

To illustrate the first step, multiply and bcc

Step 3:

The product of rows and columns is them subtracted

Step 4:

To illustrate this, ad - bc

Step 5:

The final result after subtracting the product is the value of the determinant

|A| = ad - bc


Let’s understand this further by taking a numerical example


Example

Question

Solve for det(A)

DetA = \[\begin{vmatrix}5 &7 \\ 3 & 1 \end{vmatrix}\]


Answer: Follow the below-mentioned steps to solve for the value of the determinant


Step 1: Cross multiply the rows with columns


Step 2: The product of multiplication are 5 (5 x 1) and 21(7 x 3)


Step 3: Subtraction of the products


Step 4: 5 - 21 = -16


Step 5: det(A) = -16


Therefore, following the above-mentioned step can lead to solving any determinants. Furthermore, other complex solution examples will be discussed in this article where determinant of a singular matrix and 3x3 matrix shall also be addressed.


2 x 2 Matrix Determinant

Also commonly known as a determinant of a square matrix. A 2x2 matrix has two columns and two rows. The example mentioned above is an example of a 2x2 matrix determinant. 


A matrix given below can be solved using the steps mentioned above 


det(A) = \[\begin{vmatrix}a_{11} &b_{12} \\ c_{21} & d_{22} \end{vmatrix}\]


det(A) = a11 x a22  -  a12 x a21


Using the formula above, and solve for any 2x2 determinant matrix


3x3 Matrix Determinant

A 3x3 matrix determinant has three columns and three rows. The method involving solving for the 3x3 determinant matrix is different from what has been discussed until now in this article.

 

An example of how the 3x3 matrix is represented is given below:


det (A)

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


In order to solve for a 3x3 matrix determinant, follow the steps mentioned below:


Step 1: By expanding any one row, the solution for the determinant can be derived


Step 2: For solving det(A), the first row will be expanded


Step 3: The expanded version of the determinant will be as follow:

 

a11 \[\begin{vmatrix}a_{22} &b_{23} \\ a_{32} & a_{33} \end{vmatrix}\]   -  a12\[\begin{vmatrix}a_{21} &b_{23} \\ a_{31} & a_{33} \end{vmatrix}\] + a13\[\begin{vmatrix}a_{21} &b_{22} \\ a_{31} & a_{32} \end{vmatrix}\]


Step 4: The 2x2 determinant matrix will be solved as mentioned above.


Step 5: After solving for those, simple multiplication with the items of row 1 can lead to the next and final step


Step 6: In the 3x3 determinant matrix, the signs are alternative ( +, -, +). Following these signs, one can get to the final answer of det(A). 


Let's solve an example with numerical values to get a better understanding of solving for a 3x3 determinant matrix.


Example:


Question

Solve for det(A) which is 


\[\begin{vmatrix}5 &2  &1 \\-2 &-1  &1 \\-4  &4  &3 \end{vmatrix}\]


Answer: Follow the step below in order to solve for det(A)


Step 1: For solving det(A), row 1 shall be expanded


Step 2: That being said, the expanded version of this determinant is given below


5\[\begin{vmatrix}-1 &1 \\ 4 & 3 \end{vmatrix}\]   -2\[\begin{vmatrix}-2 &1 \\ -4 & 3 \end{vmatrix}\] + 1\[\begin{vmatrix}-2 &-1 \\ -4 & 4 \end{vmatrix}\]


Step 3: This step involves solving for the 2x2 determinant matrices


5 { (-1 x 3) - (4 x 1)}  - 2 { ( -2 x 3 ) - ( 1 x -4)}  + 3 {(-2 x 4) - ( -1 x -4)}


5 ( -3 - 4) - 2 (-6 + 4) + 3( -6 -4)


5(-7) -2(-2) +3(-10)


-35 + 4 -30


-61


Step 4: The value of det(A) is - 6m1


Following the above-mentioned step can easily help you solve for 3x3 determinant matrices. It is very important to remember to expand the row as the first step and then solving as a 2 x 2 matrix.


Determinants and matrices are two different concepts but have overlap uses. Even though they can be solved using simple mathematical rules, understanding the step for the solution is important.  Similar to the 3 x 3 matrix, other square matrices can also be solved following the same steps and approaches.

FAQ (Frequently Asked Questions)

1. What is a Matrix? How can we Define the Determinant of a Matrix? Is it Compulsory for a Matrix to be Square, to find its Determinant?

Answer: Matrices can be defined as a rectangular or square arrangement of numbers or variables, in the form of columns and rows. Square matrices have an equal number of rows and columns while the rectangular matrices do not have an equal number of rows and columns. The determinant of the given matrix is its scalar value.


Yes, a matrix's determinant can only be found if the given matrix is square i. e. the number of rows is equal to the number of columns.

2. Is the Method to Find a 2 x 2 Matrix Determinant Different from the Method of Finding a 3x3 Matrix Determinant?

Answer: No, the determinant of a 2 x 2 matrix can be found in the same way as a determinant of a 3 x 3 matrix is found. A 2 x 2 matrix is the one that has two rows and two columns. While a 3 x 3 matrix is the one that has three columns and three rows. This is the basic difference between these two different matrices. Furthermore, the process of solving for a 3 x 3 determinant involves expanding one row of the matrix and solving it as a 2 x 2 determinant.