1. Definition and Examples
1. Determinant of a square matrix
If `A=[[a_11,a_12,a_13],[a_21,a_22,a_23],[a_31,a_32,a_33]]` then
`|[a_11,a_12,a_13],[a_21,a_22,a_23],[a_31,a_32,a_33]|` is called a determinant of matrix A and it is denoted by `|A|`
Determinant of `2 xx 2` matrix
`|A|=|[a,b],[c,d]| = ad - bc`
Determinant of `3 xx 3` matrix
`|A|=|[a,b,c],[d,e,f],[g,h,i]| = a|[e,f],[h,k]| - b |[d,f],[g,k]| + c |[d,e],[g,h]|`
Example
1. Find `| A |` ...
`A=[[5,6],[1,2]]`
Solution:
`=5 × 2 - 6 × 1`
`=10 -6`
`=4`
2. Find `| A |` ...
`A=[[3,1,1],[-1,2,1],[1,1,1]]`
Solution:
| `|A|` | = | | `3` | `1` | `1` | | | `-1` | `2` | `1` | | | `1` | `1` | `1` | |
|
`=3 xx (2 × 1 - 1 × 1) -1 xx (-1 × 1 - 1 × 1) +1 xx (-1 × 1 - 2 × 1)`
`=3 xx (2 -1) -1 xx (-1 -1) +1 xx (-1 -2)`
`=3 xx (1) -1 xx (-2) +1 xx (-3)`
`= 3 +2 -3`
`=2`
3. Find `| B |` ...
`B=[[2,3,1],[0,5,6],[1,1,2]]`
Solution:
| `|B|` | = | | `2` | `3` | `1` | | | `0` | `5` | `6` | | | `1` | `1` | `2` | |
|
`=2 xx (5 × 2 - 6 × 1) -3 xx (0 × 2 - 6 × 1) +1 xx (0 × 1 - 5 × 1)`
`=2 xx (10 -6) -3 xx (0 -6) +1 xx (0 -5)`
`=2 xx (4) - -3 xx (-6) +1 xx (-5)`
`= 8 +18 -5`
`=21`
This material is intended as a summary. Use your textbook for detail explanation.
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