How was this determinant calculated

Calculate determinant

The formula for calculating a 3x3 determinant is

\ (\ begin {align *} | A | = \ begin {vmatrix} a & b & c \ d & e & f \ g & h & i \ end {vmatrix} & = a \ cdot e \ cdot i + b \ cdot f \ cdot g + c \ cdot d \ cdot h \ & \ quad -g \ cdot e \ cdot c - h \ cdot f \ cdot a - i \ cdot d \ cdot b \ end {align * } \)

Examples

\ (\ begin {align *} | A | & = \ begin {vmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \ end {vmatrix} \ & = 1 \ cdot 5 \ times 9 + 2 \ times 6 \ times 7 + 3 \ times 4 \ times 8 \ & \ times 8 \ times \ times 5 \ times 3 - 8 \ times 6 \ times 1 - 9 \ times 4 \ times 2 \ & = 45 + 84 + 96 - 105 - 48 - 72 \ & = 0 \ \ end {align *} \)

\ (\ begin {align *} | B | & = \ begin {vmatrix} 2 & 5 & 2 \ 3 & -3 & 1 \ 1 & 4 & -4 \ end {vmatrix} \ & = 2 \ cdot (-3) \ cdot (-4) + 5 \ times 1 \ times 1 + 2 \ times 3 \ times 4 \ & \ quad - 1 \ times (-3) \ times 2 - 4 \ times \ times 1 \ times 2 - (-4) \ times 3 \ times 5 \ & = 24 + 5 + 24 - (-6) - 8 - (-60) \ & = 111 \ \ end {align *} \)

You can find out more about this topic in the chapter Calculating 3x3 determinants.

Calculate nxn determinant

There are essentially two methods for larger determinants, for each of which we have separate articles on offer:

In summary, it can be said that it is not that difficult to calculate determinants. Here, too, of course: Practice makes perfect!