Understanding the Problem: Determinant of a Modified Matrix

The question asks us to find the determinant of a specific matrix, which is derived from an original matrix A by rearranging its elements. We are given the original matrix A and its determinant, denoted by Δ. The new matrix has its elements \(a_{ij}\) from the original matrix A arranged in a different order.

The original matrix A is:

The determinant of matrix A is given as \(\det(A) = \Delta\).

The new matrix, let's call it B, is:

We need to find the value of \(\det(B)\) in terms of \(\Delta\).

Key Properties of Determinants

To solve this, we can use the properties of determinants related to matrix operations:

• The determinant of a matrix transpose is equal to the determinant of the original matrix: \(\det(A^T) = \det(A)\).
• Swapping two rows or two columns of a matrix changes the sign of its determinant.
• Performing multiple row or column swaps changes the sign of the determinant by \( (-1)^k \), where \(k\) is the number of swaps.

Step-by-Step Transformation

Let's see how matrix B can be obtained from matrix A using a series of transformations. It's often helpful to consider the transpose of A.

The transpose of A, denoted by \(A^T\), is obtained by swapping rows and columns of A:

We know that \(\det(A^T) = \det(A) = \Delta\).

Now let's see if we can obtain matrix B from \(A^T\) by performing column and/or row operations. Let the columns of \(A^T\) be \(C^T_1, C^T_2, C^T_3\). So, \(A^T = [C^T_1 \ C^T_2 \ C^T_3]\).

• \(C^T_1 = \begin{pmatrix} a_{11} \\ a_{12} \\ a_{13} \end{pmatrix}\)
• \(C^T_2 = \begin{pmatrix} a_{21} \\ a_{22} \\ a_{23} \end{pmatrix}\)
• \(C^T_3 = \begin{pmatrix} a_{31} \\ a_{32} \\ a_{33} \end{pmatrix}\)

Let's look at the columns of matrix B:

• Column 1 of B = \(\begin{pmatrix} a_{21} \\ a_{23} \\ a_{22} \end{pmatrix}\)
• Column 2 of B = \(\begin{pmatrix} a_{31} \\ a_{33} \\ a_{32} \end{pmatrix}\)
• Column 3 of B = \(\begin{pmatrix} a_{11} \\ a_{13} \\ a_{12} \end{pmatrix}\)

Let's try permuting the columns of \(A^T\). Consider the matrix formed by columns \(C^T_2, C^T_3, C^T_1\):

Let's call this matrix N. Matrix N is obtained from \(A^T = [C^T_1 \ C^T_2 \ C^T_3]\) by rearranging the columns in the order \(C^T_2, C^T_3, C^T_1\). The permutation of columns is (2, 3, 1) from the original order (1, 2, 3).

To get from (1, 2, 3) to (2, 3, 1), we can perform column swaps:

1. Swap column 1 and column 2 of \(A^T\): \([C^T_2 \ C^T_1 \ C^T_3]\). Determinant becomes \(-\Delta\).
2. Swap column 2 and column 3 of the new matrix (which are \(C^T_1\) and \(C^T_3\)): \([C^T_2 \ C^T_3 \ C^T_1]\). Determinant becomes \((-\Delta) \times (-1) = \Delta\).

So, the determinant of matrix N is \(\det(N) = \Delta\).

Now let's compare matrix N with matrix B:

Matrix N:

Matrix B:

We can see that matrix B is obtained from matrix N by swapping its second and third rows. Let \(R_1^N, R_2^N, R_3^N\) be the rows of N. Then the rows of B are \(R_1^N, R_3^N, R_2^N\). A single row swap changes the sign of the determinant.

Therefore, \(\det(B) = -\det(N)\).

Since \(\det(N) = \Delta\), we have \(\det(B) = -\Delta\).

Summary of Determinant Changes

We started with matrix A, \(\det(A) = \Delta\).

• Taking the transpose: \(\det(A^T) = \Delta\). (No sign change)
• Permuting columns of \(A^T\) from order (1, 2, 3) to (2, 3, 1) to get matrix N. This involves 2 column swaps (e.g., (1,2,3) → (2,1,3) → (2,3,1)). Two swaps result in a sign change of \((-1)^2 = 1\). So \(\det(N) = \det(A^T) = \Delta\).
• Swapping rows 2 and 3 of matrix N to get matrix B. One row swap results in a sign change of \((-1)^1 = -1\). So \(\det(B) = -\det(N)\).

Combining these steps:

\(\det(B) = - \det(N) = - (\det(A^T)) = - (\det(A)) = -\Delta\).

Conclusion

The value of the determinant of the given matrix is \(-\Delta\).

Revision Table: Determinant Properties

Additional Information: Cofactors and Determinants

The question mentions cofactors \(C_{11}, C_{12}, C_{13}\). The determinant of a matrix A can be calculated by expanding along any row or column using cofactors. For the first row of matrix A:

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

where \(C_{ij} = (-1)^{i+j} M_{ij}\), and \(M_{ij}\) is the determinant of the submatrix obtained by deleting the \(i\)-th row and \(j\)-th column of A.

While this formula defines how \(\Delta\) is calculated, the cofactors themselves were not needed to solve this particular problem about the determinant of the permuted matrix. The solution relied purely on how row and column operations affect the determinant value.