Paper 1, Section I, A

If $A$ is an $n$ by $n$ matrix, define its determinant $\operatorname{det} A$.

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

(i) $\operatorname{det} B$, where $B$ is obtained from $A$ by multiplying one row by $\lambda$.

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

(iii) $\operatorname{det} C$, where $C$ is obtained from $A$ by switching row $k$ and row $l(k \neq l)$.

(iv) $\operatorname{det} D$, where $D$ is obtained from $A$ by adding $\lambda$ times column $l$ to column $k$ $(k \neq l)$.

*Typos? Please submit corrections to this page on GitHub.*