Row Operations and Determinant Changes | Generated by AI
That’s a great question! Let’s address the row operations and their effect on the determinant.
📐 Achieving Row Echelon Form (REF)
You are correct that swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another are the three elementary row operations used to achieve Row Echelon Form (REF).
🛑 Effect of Row Operations on the Determinant
Your statement that the determinant of a matrix doesn’t change during these operations is partially incorrect. Only one of the three elementary row operations leaves the determinant unchanged:
1. Adding a Multiple of One Row to Another
- Operation: \(R_i \leftarrow R_i + c R_j\)
- Effect on Determinant: The determinant DOES NOT change.
- If \(\det(A) = d\), then \(\det(A’) = d\).
- Example: \(\det \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} = -2\). \(\det \begin{pmatrix} 1 & 2 \ 3-2(1) & 4-2(2) \end{pmatrix} = \det \begin{pmatrix} 1 & 2 \ 1 & 0 \end{pmatrix} = -2\).
2. Multiplying a Row by a Non-Zero Scalar
- Operation: \(R_i \leftarrow c R_i\) (where \(c \ne 0\))
- Effect on Determinant: The determinant is multiplied by the scalar \(c\).
- If \(\det(A) = d\), then \(\det(A’) = c \cdot d\).
- Example: \(\det \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} = -2\). \(\det \begin{pmatrix} 1 & 2 \ 3(2) & 4(2) \end{pmatrix} = \det \begin{pmatrix} 1 & 2 \ 6 & 8 \end{pmatrix} = 8-12 = -4\). Here, \(c=2\), and the new determinant is \(2 \cdot (-2) = -4\).
3. Swapping Two Rows
- Operation: \(R_i \leftrightarrow R_j\)
- Effect on Determinant: The determinant is multiplied by \(-1\) (it changes sign).
- If \(\det(A) = d\), then \(\det(A’) = -d\).
- Example: \(\det \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} = -2\). \(\det \begin{pmatrix} 3 & 4 \ 1 & 2 \end{pmatrix} = 3(2) - 4(1) = 6-4 = 2\). Here, the sign changed from \(-2\) to \(2\).
💡 Summary Table
| Elementary Row Operation | Notation | Effect on Determinant |
|---|---|---|
| Row Swap | \(R_i \leftrightarrow R_j\) | \(\det(A’) = -\det(A)\) |
| Scalar Multiplication | \(R_i \leftarrow c R_i\) | \(\det(A’) = c \cdot \det(A)\) |
| Row Addition | \(R_i \leftarrow R_i + c R_j\) | \(\det(A’) = \det(A)\) |
If you use row operations to transform a matrix \(A\) into its Row Echelon Form \(U\), you can find the original determinant by keeping track of the changes:
\(\)\det(A) = (-1)^s \cdot \frac{1}{c_1 c_2 \dots c_k} \cdot \det(U)\(\)
where \(s\) is the number of row swaps and \(c_1, c_2, \dots, c_k\) are the scalars used in row multiplications. For a matrix in REF, \(\det(U)\) is simply the product of its diagonal entries (since it’s an upper triangular matrix).
Would you like to try using these operations on a specific matrix to find its determinant?