Calculating a Third Order Determinant with a Formula
The question asks for the value of a determinant of the third order. The element in the i-th row and j-th column, denoted by \(a_{ij}\), is given by the formula \(a_{ij} = 2(i+j)\).
Constructing the Determinant Matrix
A third-order determinant has 3 rows and 3 columns. We can find each element using the given formula:
There are several ways to evaluate this determinant. We can expand it, or we can use properties of determinants. Let's look at the relationships between the rows:
Row 1: (4, 6, 8)
Row 2: (6, 8, 10)
Row 3: (8, 10, 12)
Observe the difference between consecutive rows:
\(R_2 - R_1 = (6-4, 8-6, 10-8) = (2, 2, 2)\)
\(R_3 - R_2 = (8-6, 10-8, 12-10) = (2, 2, 2)\)
Since the difference between consecutive rows is constant, the rows of the determinant are in Arithmetic Progression (AP).
A fundamental property of determinants states that if the rows (or columns) of a determinant are in Arithmetic Progression, the value of the determinant is zero.
Alternatively, we can use row operations to simplify the determinant. Let R1, R2, and R3 denote the first, second, and third rows, respectively.
Now observe the relationship between the second row \(R_2 = (2, 2, 2)\) and the third row \(R_3 = (4, 4, 4)\). We can see that \(R_3 = 2 \times R_2\).
Another property of determinants states that if one row (or column) is a scalar multiple of another row (or column), the value of the determinant is zero.
Since \(R_3\) is a multiple of \(R_2\), the value of the determinant is 0.
Conclusion
Using either the property of rows in AP or row operations revealing dependent rows, we find that the value of the determinant is 0.
Element
Formula \(a_{ij} = 2(i+j)\)
Value
\(a_{11}\)
2(1+1)
4
\(a_{12}\)
2(1+2)
6
\(a_{13}\)
2(1+3)
8
\(a_{21}\)
2(2+1)
6
\(a_{22}\)
2(2+2)
8
\(a_{23}\)
2(2+3)
10
\(a_{31}\)
2(3+1)
8
\(a_{32}\)
2(3+2)
10
\(a_{33}\)
2(3+3)
12
Revision Table: Determinants and Properties
Concept
Description
Relevance to Problem
Third Order Determinant
A square array of numbers (3x3) whose value is calculated by a specific rule.
The problem involves calculating the value of a 3x3 determinant.
Element \(a_{ij}\)
The number in the i-th row and j-th column of the determinant/matrix.
The elements are defined by the formula \(a_{ij} = 2(i+j)\).
Determinant with Rows/Columns in AP
If rows or columns form an Arithmetic Progression (constant difference between consecutive rows/columns), the determinant value is zero.
The rows (and columns) of the calculated determinant are in AP.
Determinant with Linearly Dependent Rows/Columns
If one row/column can be expressed as a linear combination of other rows/columns (e.g., one row is a multiple of another), the determinant value is zero.
Row operations showed \(R_3 = 2 R_2\), indicating linear dependence.
Additional Information: Properties of Determinants
Understanding the properties of determinants is crucial for solving many linear algebra problems quickly. Some key properties include:
Transpose: The determinant of the transpose of a matrix is equal to the determinant of the original matrix (\(\det(A^T) = \det(A)\)).
Row/Column Swap: Swapping two rows or two columns of a determinant changes its sign.
Scalar Multiplication: Multiplying a single row or column by a scalar \(k\) multiplies the determinant value by \(k\). Multiplying an entire \(n \times n\) determinant by \(k\) results in a determinant value multiplied by \(k^n\).
Row/Column Operations (Elementary):
Adding a multiple of one row (or column) to another row (or column) does not change the value of the determinant. This is what we used in the solution (\(R_i \to R_i + k R_j\)).
Multiplying a row/column by a non-zero scalar \(k\) multiplies the determinant by \(k\).
Zero Row/Column: If a determinant has a row or a column consisting entirely of zeros, its value is zero.
Identical Rows/Columns: If a determinant has two identical rows or two identical columns, its value is zero. Our example with \(R_3 = 2 R_2\) is a case of linearly dependent rows, which also results in a zero determinant.
Determinant of Product: The determinant of the product of two square matrices is the product of their determinants (\(\det(AB) = \det(A) \det(B)\)).
Recognizing patterns like rows/columns in Arithmetic Progression or linear dependency can significantly simplify the calculation of determinants.
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