Understanding Determinants and Cofactors

The question asks for the value of a specific expression involving elements of a matrix and the cofactors of elements from a different row. Let's first understand the key terms: determinant, matrix, and cofactors.

What is a Matrix?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The given matrix A is a 3x3 matrix:

\(A = \left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right)\)

What is a Determinant?

The determinant (\(\Delta\)) is a scalar value that can be computed from the elements of a square matrix. The determinant provides important information about the matrix, such as whether it is invertible.

What are Cofactors?

The cofactor \(C_{ij}\) of an element \(a_{ij}\) in a matrix is calculated as \(C_{ij} = (-1)^{i+j} M_{ij}\), where \(M_{ij}\) is the minor of \(a_{ij}\). The minor \(M_{ij}\) is the determinant of the submatrix obtained by deleting the i-th row and j-th column of the original matrix.

In this question, \(C_{11}, C_{12}, C_{13}\) are the cofactors of the elements in the first row (\(a_{11}, a_{12}, a_{13}\)) respectively.

Determinant Expansion Using Cofactors

The determinant of a matrix can be calculated by expanding along any row or any column. The expansion along the i-th row is given by:

\(\Delta = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3}\)

For example, expanding along the first row (\(i=1\)):

\(\Delta = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\)

Expanding along the second row (\(i=2\)):

\(\Delta = a_{21}C_{21} + a_{22}C_{22} + a_{23}C_{23}\)

And expanding along the third row (\(i=3\)):

\(\Delta = a_{31}C_{31} + a_{32}C_{32} + a_{33}C_{33}\)

Key Property: Elements of One Row with Cofactors of Another Row

There is a fundamental property of determinants that states: the sum of the products of the elements of any row (or column) with the cofactors of the corresponding elements of a different row (or column) is always zero.

Mathematically, for a matrix A, the sum of the products of elements of the i-th row and the cofactors of the j-th row is:

\(a_{i1}C_{j1} + a_{i2}C_{j2} + a_{i3}C_{j3}\)

This sum is equal to:

• \(\Delta\) if \(i = j\) (expansion along the i-th row)
• \(0\) if \(i \neq j\)

Evaluating the Given Expression

The expression we need to evaluate is \(a_{21} C_{11} + a_{22} C_{12} + a_{23} C_{13}\).

Let's compare this expression to the general form \(a_{i1}C_{j1} + a_{i2}C_{j2} + a_{i3}C_{j3}\):

• The elements \(a_{21}, a_{22}, a_{23}\) are from the second row, so \(i=2\).
• The cofactors \(C_{11}, C_{12}, C_{13}\) are of the elements from the first row, so \(j=1\).

In this case, we have \(i=2\) and \(j=1\). Since \(i \neq j\), the sum of the products \(a_{21} C_{11} + a_{22} C_{12} + a_{23} C_{13}\) must be zero based on the property mentioned above.

This property holds true because essentially, multiplying elements of one row by cofactors of another row corresponds to calculating the determinant of a matrix where that 'other' row has been replaced by the 'one' row. A matrix with two identical rows has a determinant of zero.

Conclusion

Based on the property that the sum of the products of the elements of one row and the cofactors of another row is zero, the value of \(a_{21} C_{11} + a_{22} C_{12} + a_{23} C_{13}\) is 0.

The expression \(a_{21} C_{11} + a_{22} C_{12} + a_{23} C_{13}\) represents the sum of the products of the elements of the second row (\(a_{21}, a_{22}, a_{23}\)) and the cofactors of the elements of the first row (\(C_{11}, C_{12}, C_{13}\)). Since the rows involved are different (row 2 and row 1), the value of this sum is 0.

Revision Table: Matrix Determinant and Cofactors

Additional Information: Applications of Determinants

Determinants are not just theoretical concepts; they have various applications in mathematics and other fields. Some important applications include:

• Solving Systems of Linear Equations: Using Cramer's rule, determinants can be used to find the unique solution to a system of linear equations.
• Finding the Inverse of a Matrix: A square matrix has an inverse if and only if its determinant is non-zero. The inverse can be calculated using the adjoint matrix, which involves cofactors.
• Calculating Area and Volume: The absolute value of the determinant of a matrix formed by vectors can represent the area of a parallelogram (in 2D) or the volume of a parallelepiped (in 3D) spanned by these vectors.
• Eigenvalues: Determinants are used in finding the eigenvalues of a matrix by solving the characteristic equation \(\det(A - \lambda I) = 0\).
• Linear Transformations: The determinant of the matrix representing a linear transformation indicates how the transformation scales area or volume.

Understanding determinants and their properties, like the one used in this problem, is fundamental in linear algebra.