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The DeterminantÂļ

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The formula \(|\mathbf{C}| = |\mathbf{P}|^{n-1}\) is a property of a cofactor matrix (C) derived from an original square matrix (P) of order \(n \times n\). Specifically, the determinant of the cofactor matrix is equal to the determinant of the original matrix raised to the power of \(n - 1\).

Explanation of the Formula

  • \(|\mathbf{C}|\) is the determinant of the cofactor matrix.
  • \(|\mathbf{P}|\) is the determinant of the original square matrix \(\mathbf{P}\).
  • \(n\) is the order (dimension) of the square matrix (e.g., \(n = 2\) for a \(2 \times 2\) matrix, \(n = 3\) for a \(3 \times 3\) matrix).

Example: \(2 \times 2\) Matrix (\(n = 2\))

Let \(\mathbf{P}\) be a \(2 \times 2\) matrix:

\[\mathbf{P} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

Step 1: Find the determinant of \(\mathbf{P}\) (\(|\mathbf{P}|\) )
The determinant of a \(2 \times 2\) matrix is given by:

\[|\mathbf{P}| = ad - bc\]

Step 2: Find the cofactor matrix \(\mathbf{C}\) of \(\mathbf{P}\)
The cofactor \(C_{ij}\) of an element \(a_{ij}\) is given by \((-1)^{i+j} \times M_{ij}\), where \(M_{ij}\) is the minor.

  • Cofactor of \(a\) (Cofactor of \(P_{11}\)): \((-1)^{1+1} \times M_{11} = 1 \times d = d\)
  • Cofactor of \(b\) (Cofactor of \(P_{12}\)): \((-1)^{1+2} \times M_{12} = -1 \times c = -c\)
  • Cofactor of \(c\) (Cofactor of \(P_{21}\)): \((-1)^{2+1} \times M_{21} = -1 \times b = -b\)
  • Cofactor of \(d\) (Cofactor of \(P_{22}\)): \((-1)^{2+2} \times M_{22} = 1 \times a = a\)

The cofactor matrix is:

\[\mathbf{C} = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}\]

Step 3: Find the determinant of \(\mathbf{C}\) (\(|\mathbf{C}|\) )

\[|\mathbf{C}| = (d)(a) - (-c)(-b) = ad - bc\]

Step 4: Verify the formula \(|\mathbf{C}| = |\mathbf{P}|^{n-1}\)
For \(n = 2\), the formula is \(|\mathbf{C}| = |\mathbf{P}|^{2-1} = |\mathbf{P}|^{1} = |\mathbf{P}|\).

We found that \(|\mathbf{C}| = ad - bc\) and \(|\mathbf{P}| = ad - bc\). Thus, the formula holds:

\[ad - bc = (ad - bc)^1\]

Example with numbers

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  1. \(|\mathbf{P}|: (2 \times 4) - (3 \times 1) = 8 - 3 = 5\)

  2. \(\mathbf{C}\): The cofactor matrix is \(\begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix}\)

  3. \(|\mathbf{C}|: (4 \times 2) - (-1 \times -3) = 8 - 3 = 5\)

  4. Verification: \(|\mathbf{C}| = 5\), \(|\mathbf{P}|^{n-1} = 5^{2-1} = 5^1 = 5\). The formula \(5 = 5\) is correct.