Linear Mathematics | Part IB, 2002

Define the determinant of an n×nn \times n matrix AA, and prove from your definition that if AA^{\prime} is obtained from AA by an elementary row operation (i.e. by adding a scalar multiple of the ii th row of AA to the jj th row, for some jij \neq i ), then detA=detA\operatorname{det} A^{\prime}=\operatorname{det} A.

Prove also that if XX is a 2n×2n2 n \times 2 n matrix of the form

(ABOC)\left(\begin{array}{ll} A & B \\ O & C \end{array}\right)

where OO denotes the n×nn \times n zero matrix, then detX=detA\operatorname{det} X=\operatorname{det} A det CC. Explain briefly how the 2n×2n2 n \times 2 n matrix

(BIOA)\left(\begin{array}{ll} B & I \\ O & A \end{array}\right)

can be transformed into the matrix

(BIABO)\left(\begin{array}{cc} B & I \\ -A B & O \end{array}\right)

by a sequence of elementary row operations. Hence or otherwise prove that detAB=\operatorname{det} A B= detAdetB\operatorname{det} A \operatorname{det} B.

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