How to Calculate the Determinant of a Matrix?

In the previous articles, we have seen what a matrix is, its types(square matrix, diagonal matrix, identity matrix, symmetric matrix, skew-symmetric matrix and many more). Now, it’s time to level up the learning. This article will discuss how you can find the determinant of a matrix.

What is Determinant of a Matrix?

The determinant of a matrix is a scalar value that is calculated from the elements of the Square matrix. It is used to determine whether a given matrix is invertible or not, to solve a system of linear algebra, to find the eigenvalue of a matrix, and for geometrical interpretations.

If A is any matrix, then the determinant of the matrix is given by: det(A) or |A|.

Example:

If A = 

Then the determinant of A is:

det(A) = |A| = -10.

Properties of Determinant

• The determinant of a matrix is zero if a row, or a column of a matrix is all zeros.
• Determinant of a matrix is equal to the product of its eigenvalues.
• Determinant of an inverse matrix is equal to the reciprocal of the determinant of the original matrix.
• Determinant of the product of two matrices is equal to the product of the determinant of the individual matrix.

Want to learn, how to find the determinant of a matrix, check out the video

Until now, you have a clear understanding of what the determinant of a matrix is, and its properties. Now, it's time to see how to calculate the determinant of the 2x2, 3x3, and 4x4 matrix.

Determinant of 2x2 Matrix

Let B =

Then the determinant of A will be:

det(B) = |B| = ad - bc

Now, let’s have an example to get a better understanding of how to calculate the determinant of a 2x2 matrix.

Example

C = 

Here, a = 3, b = 8, c = 4, and d = 6

Now, substituting the values in the above formula, we will get:

det (C) = |C| = ad - bc = 3*6 - 8*4 = 18 - 32 = -14

=> det (C) = |C| = - 14

Now, before moving forward let’s discuss the concepts of Minors and Cofactors, that will be used to calculate the determinant of 3x3 and higher order matrices.

Minor

A minor of a matrix is the determinant of the smaller matrix, that you get by removing the row and column of that element.

Cofactor

The cofactor of the elements is calculated by (-1)^{(i + j)} times its minor, where i and j are the corresponding row and column of that element.

Didn’t get the concept of minor and cofactor. Don’t worry, in the next section we will see how to calculate these.

Determinant of 3x3 matrix

Let D be a 3x3 matrix

Then, the determinant of the matrix is calculated using the following steps:

• Choose a row or column (typically we select the first row or first column)
• Calculate the cofactor for each element in the chosen row.
• For example, the cofactor of a in the matrix D

• Multiply each element with their corresponding cofactor.
• Finally, add these products to get the determinant.

Now, it’s time for an example.

Let, E

To calculate the determinant of the above matrix, we will choose the first row (2, 3, 4) to find the cofactor of the matrix.

Calculate the cofactor

Multiply and Sum

2 * (-3) + 3 * (-6) + 4 * (-3) = (-6) + (-18) +  (- 12 ) = -36.

Hence, the determinant of the matrix E is -36.

Determinant of 4x4 Matrix

To calculate the determinant of a 4x4 matrix, you just have to follow the same step as in the above 3x3 matrix. Here the step will be more complex, as you have to calculate the minor of the 3x3 matrix and for 4 elements.

So, here are two tips and tricks that you can follow, to calculate the determinant of a 4x4 matrix easily and quickly.

• Choose a row or a column with having zero, this will make your calculation easy, as the product with zero is always a zero.
• Focus on one minor at a time.

Here is one question for your practice.

Find the determinant of the given 4x4 matrix.

Role of Determinant in Solving System of Linear Equations

• If the determinant of a matrix is zero, then there doesn’t exist any unique solution.
• If the determinant of the matrix is non-zero, then there exists a unique solution and the system is said to be consistent and independent. 

Conclusion

The determinant of a matrix is not just a number, it has application across different domains like engineering, physics, and computer science. The most common application includes finding the solution of the system of linear equations and finding the inverse of any matrix.

This article tries to give you the complete and in-depth understanding of what and how to calculate the determinant of an article.

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