58. Find the determinant of the following matrix:
| 1 3 0 |
| 2 6 4 |
| −1 −1 2 |
Determinant of a 3×3 Matrix Using Cofactor Expansion
Among the various techniques available for evaluating determinants, cofactor expansion is one of the most systematic and reliable methods. Although shortcut techniques exist for specific matrices, mastering cofactor expansion enables students to solve any determinant problem confidently. In this solution, we calculate the determinant of the given 3×3 matrix step by step while explaining the mathematical reasoning behind every operation.
Given Matrix
The matrix is
A =
| 1 3 0 |
| 2 6 4 |
| −1 −1 2 |
Our objective is to determine the value of |A|, which represents the determinant of the matrix.
Concept Behind the Question
The determinant is a numerical value associated with every square matrix. It provides important information about the properties of the matrix. For example, if the determinant is zero, the matrix is singular and cannot be inverted. If the determinant is non-zero, the matrix is invertible and has full rank.
For a 3×3 matrix, one of the most widely used approaches is the cofactor expansion method. In this method, we expand the determinant along any row or column. The first row is generally preferred when it contains one or more zero elements because this reduces the amount of computation. In the present matrix, the first row contains a zero in the third position, making cofactor expansion along the first row the simplest approach.
Step 1: Write the Cofactor Expansion Formula
Expanding the determinant along the first row gives
|A| = 1(M11) − 3(M12) + 0(M13)
where each M represents the determinant of the corresponding 2×2 minor matrix.
Since the third element of the first row is zero, its contribution automatically becomes zero. Therefore, only two minor determinants need to be evaluated.
Step 2: Calculate the First Minor
After removing the first row and first column, the remaining 2×2 matrix is
| 6 4 |
| −1 2 |
The determinant of a 2×2 matrix is calculated as
(ad − bc)
Therefore,
=(6 × 2) − (4 × −1)
=12 + 4
=16
Step 3: Calculate the Second Minor
Next, remove the first row and second column.
The resulting 2×2 matrix is
| 2 4 |
| −1 2 |
Its determinant becomes
=(2 × 2) − (4 × −1)
=4 + 4
=8
Step 4: Substitute the Values
Substituting the calculated minors into the cofactor expansion formula gives
|A| = (1 × 16) − (3 × 8) + (0 × anything)
=16 − 24
= −8
Why Was the First Row Chosen for Expansion?
Although a determinant can be expanded along any row or column, choosing the row or column containing the maximum number of zeros reduces the amount of calculation. Since the first row contains a zero, one complete minor determinant is eliminated automatically, making the computation shorter and less prone to arithmetic errors.
In competitive examinations where speed is important, selecting the appropriate row or column for expansion can significantly reduce solving time.
Physical and Mathematical Significance of Determinants
Determinants play a crucial role in many branches of mathematics, physics, engineering, and data science. They are used to determine whether a system of linear equations has a unique solution, whether vectors are linearly independent, and whether a matrix possesses an inverse.
In geometry, the absolute value of a determinant represents area or volume scaling after a linear transformation. A determinant equal to zero indicates that the transformation compresses space into a lower dimension, making the matrix singular.
Key Takeaways
Choose the Simplest Row or Column
Select the row or column containing the greatest number of zero elements whenever possible. This minimizes calculations and improves speed.
Remember the Alternating Sign Pattern
The cofactor expansion follows the alternating sign sequence (+, −, +) in the first row. Ignoring this sign convention is one of the most common sources of mistakes during determinant calculations.
Evaluate Each Minor Carefully
Each minor is simply the determinant of the remaining 2×2 matrix after removing the corresponding row and column. Accurate evaluation of these minors is essential for obtaining the correct determinant.
A Non-Zero Determinant Indicates an Invertible Matrix
Since the determinant of the given matrix is −8, which is not equal to zero, the matrix is non-singular and possesses an inverse.
Conclusion
This problem demonstrates the effectiveness of the cofactor expansion method for evaluating determinants of 3×3 matrices. By selecting the first row for expansion, calculating the required 2×2 minors, and carefully applying the alternating sign convention, the determinant is obtained as −8.
Final Answer
The determinant of the given matrix is
−8
