Step-by-Step Guide to Calculating the Determinant of a 3x3 Matrix
Definition of Determinant of a 3 x 3 Matrix
In mathematics, a determinant is a value that is described for a square matrix. It is of crucial importance when solving systems of linear equations using a matrix. Determinants are the special numbers in matrices. Determinants are calculated from the square matrix. Likewise, the determinant of a 3 x 3 matrix is computed for a matrix with 3 rows and 3 columns, implying that the matrix must have an equal number of rows and columns. Those beings, so, let’s understand what the determinant of a matrix is. Imparting the knowledge that a matrix is an arrangement accommodating information of a linear transformation, and that this arrangement can comply with the coefficients of each variable in an equation system, we can literally describe the function of a determinant.
Mathematically, the determinant of 3x3 matrices is defined as
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Importance of Determinant in Linear Transformation
A determinant is all capable of scaling the linear transformation from the matrix. It can help you to attain the inverse of the matrix (given that there is one) and will further allow solving the systems of linear equations by initiating circumstances in which we can anticipate certain outcomes or features from the system (dependent on the determinant and the type of linear system). It also allows us to have familiarity with the fact if we may anticipate a unique solution, number of solutions (one or more than one) or none at all for the system.
Finding Determinant of a 3x3 Matrix
Typically, there are 2 methods of assessing the determinant of a 3x3 matrix to employ as following
General Method
In order to obtain the determinant of a 3x3 matrix using the general method, break down the matrix into secondary matrices of shorter dimensions in a procedure referred to "expansion of the first row".
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Shortcut Method
This is a clever trick to obtain the determinant of a 3x3 matrix that equips the calculation of a determinant of a large matrix by straightaway multiplying and subtracting or (adding) all of the information elements in their relevant module, without having to go all across the matrix expansion of the first row as well assessing the determinants of secondary matrices'.
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Uses of Determinant of a Matrix
The determinant of a matrix is most commonly used in calculus, advanced geometry and linear algebraic systems. Finding the determinant of a matrix becomes easier with few practice sessions. So, let’s get going with some practice problems.
Solved Examples
How to Find the Determinant of a 3×3 Matrices
Problem 1: Find the determinant of the 3×3 matrix below
Let’s say M = [2 -3 1]
[2 0 -1]
[1 4 5]
Solution 1:
In order to find out the determinant of the 3×3 matrix
We create here a set-up to enable you establish the correspondence between the generic elements of the formula and the elements of the real problem
[M N O] [2,-3, 1]
[P Q R] = 3 by 3 matrix with elements [2, 0,-1]
[S T U] [1, 4, 5]
Now, using the formula to find the determinant of a square matrix
[M N O]
det. [P Q R] = m. det [Q R; T U] - n. det [P R; S U] + O. det [P Q; S T]
[S T U]
det [2,-3, 1; 2, 0,-1;1, 4, 5] = 2. det [ 0 -1; 4 5] - (-3) .det [ 2 -1; 15] + 1.det [2 0; 1 4]
Thus, we obtain
= 2 [0- [ -4} ] + 3 [ 10 - {-1}] + 1 [8-0]
= 2 {0+4} + 3 {10+1} +1 {8}
= 2 {4} + 3 {11} + {8}
Hence,
= 8 + 33 + 8
= 49
Problem 2:
Find out the matrix P as described below:
P= [2 -5 3; 0 7 -2; -1 4 1]
Solution 2:
Using the shortcut method gives us
det | P| = [ { 2 × 7 × 1) + { -5× -2 × -1} + { 3×0×4} - [{3 × 7 × -1} + { 2 × -2 × 4} +
{-5 ×0 × 1}]
Thus, we obtain
det | P| = ( 14 - 10 + 0) - ( 21 - 16 + 0) = 4 - (-37) = 41
Fun Facts
There is a condition to attain a matrix determinant that is that the matrix should be a square matrix in order to compute it.
There is no existence of the determinant of a non square matrix.
Only determinants of square matrices are described by way of maths
We generally write down matrices and their determinants in a similar way. For example the determinant of a matrix can be simply described as det P, det (P) or |P|
FAQs on Determinant of a 3x3 Matrix: Concepts & Solutions
1. What exactly is the determinant of a 3x3 matrix?
The determinant of a 3x3 matrix is a unique scalar value calculated from its elements. This single number provides important information about the matrix, such as whether it is invertible and its properties related to linear transformations. Geometrically, it can represent the volume scaling factor of a parallelepiped defined by the matrix's column or row vectors.
2. What are minors and cofactors, and how are they used to find the determinant of a 3x3 matrix?
Minors and cofactors are essential for calculating determinants of 3x3 or larger matrices. Here’s a breakdown:
- Minor: The minor of an element (let's say 'a') is the determinant of the smaller 2x2 matrix that remains after deleting the row and column containing that element.
- Cofactor: The cofactor of an element is its minor multiplied by either +1 or -1, based on its position. The sign is determined by the formula (-1)i+j, where 'i' is the row number and 'j' is the column number.
The determinant is the sum of the products of the elements of any one row or column with their corresponding cofactors.
3. What is the significance of the determinant being zero for a 3x3 matrix?
If the determinant of a 3x3 matrix is zero, it has several critical implications as per the CBSE syllabus for 2025-26:
- The matrix is called a singular matrix.
- The matrix is not invertible, meaning an inverse matrix does not exist.
- The row and column vectors of the matrix are linearly dependent, meaning one row/column can be expressed as a combination of the others.
- When used to represent a system of linear equations, a zero determinant indicates that the system either has no solution or infinitely many solutions.
4. What happens to the value of a determinant if two rows or two columns of a 3x3 matrix are interchanged?
This is a fundamental property of determinants. If any two rows (or any two columns) of a 3x3 matrix are interchanged, the sign of its determinant changes. For example, if the determinant of matrix A is 'k', and we create a new matrix B by swapping Row 1 and Row 2 of A, then the determinant of B will be -k. This property is crucial for simplifying determinant calculations.
5. What is the relationship between the determinant of a 3x3 matrix and its inverse?
The determinant is fundamentally linked to the existence of a matrix's inverse. A 3x3 matrix 'A' has an inverse (A⁻¹) if and only if its determinant is non-zero. The formula for the inverse directly involves the determinant:
A⁻¹ = (1/det(A)) * adj(A)
where adj(A) is the adjoint of the matrix. This shows that if det(A) = 0, the expression for the inverse would involve division by zero, which is undefined. Therefore, a non-zero determinant is a prerequisite for a matrix to be invertible.
