This calculator performs three main types of operations essential for manipulating matrices: row switching, row scaling, and row addition. Each operation serves a unique purpose, facilitating users in solving linear equations and understanding matrix behaviors.
Formula of Elementary Row Operation Calculator
Row Switching (Interchanging):
- Formula: R_i ↔ R_j
- Variables: R_i and R_j represent the rows being interchanged. This operation swaps two rows in a matrix, often used to position a non-zero element in a pivotal position during Gaussian elimination.
Row Scaling (Multiplication):
- Formula: k * R_i
- Variables: k is the scalar multiplier, and R_i is the row being scaled. Scaling a row by a non-zero scalar is crucial when normalizing rows or simplifying equations.
Row Addition (Combination):
- Formula: R_i + k * R_j
- Variables: R_i is the row being modified, R_j is the row being added to R_i, and k is the scalar multiplier for R_j. Row addition combines two rows, a key step in eliminating variables to achieve row echelon form.
Helpful Tables and Conversions
Operation | Formula | Description |
---|---|---|
Row Switching | R_i ↔ R_j | Interchanges two rows i and j in a matrix. |
Row Scaling | k * R_i | Multiplies row i by a non-zero scalar k. |
Row Addition | R_i + k * R_j | Adds a multiple of row j to row i. |
Make Leading One | \frac{1}{a_{ii}} * R_i | Scales row i to make the diagonal entry a leading one. |
Zero Below Pivot | R_j – m * R_i | Subtracts m times row i from row j to create zeros below pivot. |
Examples of Elementary Row Operation Calculator
Initial Matrix
[1 2 3]
[4 5 6]
[7 8 9]
Goal
Transform the matrix into an upper triangular form using elementary row operations.
Step-by-Step Operations
- Row Switching (Interchanging) R₁ and R₃:
[7 8 9]
[4 5 6]
[1 2 3]
Row Scaling R₂ by 0.5:
[7 8 9]
[2 2.5 3]
[1 2 3]
Row Addition: Subtract 2 times R₃ from R₂:
[7 8 9]
[0 -1.5 -3]
[1 2 3]
Final Adjustment: Subtract 1/7 of R₁ from R₃:
[7 8 9]
[0 -1.5 -3]
[0 0.857 1.714]
Most Common FAQs
Opt for batch operations where possible, and use the conversion tools to manage different matrix sizes effectively.
Absolutely! It’s designed to be an educational tool to help students understand and perform matrix operations independently.