(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let A be an n x n matrix and [itex]\alpha[/itex] a scalar. Show that [itex]det(\alpha A) = \alpha^{n}det(A)[/itex]

2. Relevant equations

[itex] det(A) = a_{11}A_{11} + a_{12}A_{12} + \cdots + a_{1n}A_{1n} [/itex]

where [itex] A_{ij} = (-1)^{i+j}det(M_{ij}) [/itex]

3. The attempt at a solution

[itex] det(A) = a_{11}A_{11} + a_{12}A_{12} + \cdots + a_{1n}A_{1n} [/itex]

[itex] det(\alpha A) = \alpha a_{11}A^{\alpha}_{11} + \alpha a_{12}A^{\alpha}_{12} + \cdots + \alpha a_{1n}A^{\alpha}_{1n} [/itex]

[itex] det(\alpha A) = \alpha (a_{11}A^{\alpha}_{11} + a_{12}A^{\alpha}_{12} + \cdots + a_{1n}A^{\alpha}_{1n}) [/itex]

I can see that as I go through and calculate the cofactors I will continue to get an additional alpha coefficient each time, so that I will end up with [itex]det(\alpha A) = \alpha^{n}det(A)[/itex], but I am having trouble formalizing it. Thank you in advance for any help.

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# Homework Help: Proof for determinant of a scalar multiplied by a vector

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