Paper 1, Section I, A

Vectors and Matrices | Part IA, 2019

If AA is an nn by nn matrix, define its determinant detA\operatorname{det} A.

Find the following in terms of detA\operatorname{det} A and a scalar λ\lambda, clearly showing your argument:

(i) detB\operatorname{det} B, where BB is obtained from AA by multiplying one row by λ\lambda.

(ii) det(λA)\operatorname{det}(\lambda A).

(iii) detC\operatorname{det} C, where CC is obtained from AA by switching row kk and row l(kl)l(k \neq l).

(iv) detD\operatorname{det} D, where DD is obtained from AA by adding λ\lambda times column ll to column kk (kl)(k \neq l).

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