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Home > Matrix & Vector calculators > Determinants using Sarrus Rule example
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20. determinants using Sarrus Rule example
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- Example `[[1,2,3],[4,5,6],[7,8,9]]`
- Example `[[3,2,4],[2,0,2],[4,2,3]]`
- Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`
- Example `[[1,2,3],[0,1,0],[2,3,1]]`
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Other related methods
- Transforming matrix to Row Echelon Form
- Transforming matrix to Reduced Row Echelon Form
- Rank of matrix
- Characteristic polynomial of matrix
- Eigenvalues
- Eigenvectors (Eigenspace)
- Triangular Matrix
- LU decomposition using Gauss Elimination method of matrix
- LU decomposition using Doolittle's method of matrix
- LU decomposition using Crout's method of matrix
- Diagonal Matrix
- Cholesky Decomposition
- QR Decomposition (Gram Schmidt Method)
- QR Decomposition (Householder Method)
- LQ Decomposition
- Pivots
- Singular Value Decomposition (SVD)
- Moore-Penrose Pseudoinverse
- Power Method for dominant eigenvalue
- determinants using Sarrus Rule
- determinants using properties of determinants
- Row Space
- Column Space
- Null Space
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2. Example `[[3,2,4],[2,0,2],[4,2,3]]` (Previous example) | 4. Example `[[1,2,3],[0,1,0],[2,3,1]]` (Next example) |
3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`
Find determinants using Sarrus Rule ... `[[1,1,1],[-1,-3,-3],[2,4,4]]`
Solution:
Write first 2 columns of matrix to right of 3rd column, so we have total 5 columns.
`A=` | | 1 | 1 | 1 | 1 | 1 | | | -1 | -3 | -3 | -1 | -3 | | | 2 | 4 | 4 | 2 | 4 | |
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`A=` | | 1 | 1 | 1 | 1 | 1 | | | -1 | -3 | -3 | -1 | -3 | | | 2 | 4 | 4 | 2 | 4 | |
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Now, add products of diagonals going from top to bottom (blue lines) and subtract products of diagonals going from bottom to top (red lines).
`={1*(-3)*4+1*(-3)*2+1*(-1)*4}-{2*(-3)*1+4*(-3)*1+4*(-1)*1}`
`=((-12)+(-6)+(-4))-((-6)+(-12)+(-4))`
`=-22-(-22)`
`=0`
Method-2: Determinant by expanding cofactors`|A|` | = | | `1` | `1` | `1` | | | `-1` | `-3` | `-3` | | | `2` | `4` | `4` | |
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`=1 xx (-3 × 4 - (-3) × 4) -1 xx (-1 × 4 - (-3) × 2) +1 xx (-1 × 4 - (-3) × 2)` `=1 xx (-12 +12) -1 xx (-4 +6) +1 xx (-4 +6)` `=1 xx (0) -1 xx (2) +1 xx (2)` `= 0 -2 +2` `=0`
This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then
2. Example `[[3,2,4],[2,0,2],[4,2,3]]` (Previous example) | 4. Example `[[1,2,3],[0,1,0],[2,3,1]]` (Next example) |
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