Theorem. The determinant of the product of two matrices of order n is the product of their determinants. That is, |AB| = |A| |B|.

Example. Let

then

|AB| = |A| . |B| =   = (3) (3) = 9.

Remark. Since we known that for any square matrix A, |A| = |A’|, we have |AB| = |A| |B| = |A’| |B’| =  |A| |B’|. Thus, the product of two determinants can be written in terms of row by column as in matrices or unlike matrices column by column, column by row or row by row.

Example. Show that



Hence evaluate the determinant on the right.

Solution. Let







Further,



Δ² = 4a² b² c²

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