Solution
= Solution
The <Leibniz formula for determinants> is
$$
\boxed{\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)
\prod_{r=1}^n A_{r,\sigma(r)}}.
$$
Equivalently, the <determinant> is the unique alternating function of the rows that is <multilinearity of the determinant>[multilinear] and takes value one on the <identity matrix>. Multiplying one row by $\lambda$ therefore gives
$$
\boxed{\det B=\lambda\det A}.
$$