Understanding Matrix Determinants and Cofactors

The question asks about the value of a specific expression involving the elements of the first row of a matrix A and their corresponding cofactors. This expression is directly related to how the determinant of a matrix is calculated.

What is a Matrix Determinant?

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

For a \(3 \times 3\) matrix like the given matrix A:

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

The determinant is denoted by \(\Delta\) or det(A).

What are Cofactors?

A cofactor \(C_{ij}\) of an element \(a_{ij}\) in a matrix is defined 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 \(a_{11}, a_{12}, a_{13}\) respectively.

• \(C_{11}\) is the cofactor of \(a_{11}\).
• \(C_{12}\) is the cofactor of \(a_{12}\).
• \(C_{13}\) is the cofactor of \(a_{13}\).

Determinant Expansion using Cofactors

One of the standard methods to calculate the determinant of a matrix is by expanding along a row or a column. The determinant \(\Delta\) of a matrix A can be calculated by summing the products of the elements of any row (or column) with their corresponding cofactors.

Expanding along the first row, the determinant \(\Delta\) is given by:

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

Expanding along the second row:

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

Expanding along the first column:

\(\Delta = a_{11}C_{11} + a_{21}C_{21} + a_{31}C_{31}\)

And so on for any row or column.

Evaluating the Expression

The question asks for the value of the expression \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\). Comparing this expression with the determinant expansion formula along the first row, we can see that this expression is exactly the definition of the determinant \(\Delta\) when expanded along the first row.

Therefore, the value of \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\) is \(\Delta\).

Summary of Calculation

The sum \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\) is the formula for calculating the determinant \(\Delta\) by expanding along the first row.

Conclusion

Based on the definition of determinant expansion using cofactors, the value of \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\) for matrix A is its determinant, \(\Delta\).

Revision Table: Matrix Determinant Concepts

Additional Information: Properties of Determinants

• The determinant of a matrix is non-zero if and only if the matrix is invertible (non-singular).
• The determinant of the transpose of a matrix is equal to the determinant of the original matrix (\(\text{det}(A^T) = \text{det}(A)\)).
• If a matrix has a row or a column of zeros, its determinant is zero.
• If a matrix has two identical rows or columns, its determinant is zero.
• If a row (or column) is multiplied by a scalar k, the determinant is multiplied by k.
• If two rows (or columns) are swapped, the determinant changes sign.
• Adding a multiple of one row (or column) to another row (or column) does not change the determinant.