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next stroking out the second row and first column
and finally stroking out the third row and first column
The determinant of the 3 by 3 matrix is formed from these three 2 by 2 determinants as
This is what you have as the value of the determinant except that you forgot the leading a, d and g. The determinant can also be evaluated by expanding down the second column or down the third column, or in fact across the first, second or third rows. You need only be careful of the signs of the leading coefficients. For example, expanding across the second row gives
One of the advantages of this technique is that it extends to larger determinants. For example you can evaluate a 4 by 4 determinant by expanding down the first column and evaluating the four resulting 3 by 3 determinants. The second method you give for evaluating a 3 by 3 determinant is valid but it has a serious flaw. The method doesn't extend to larger determinants. Harley  


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