[Math] Matrices – Find the value(s) of constant k

Question

Find the values of the constant $k$ such that $(k$A$)^T(k$A$) = 28$, where:

$$A = \begin{bmatrix}-1\\2\\-3\end{bmatrix}$$

Actually, I got no idea how to solve this.
how do i solve this?
Can you please offer your assistance? Thank you

Answer 1

When you take the transpose of a vector multiplying the vector itself, you are dotting the two vectors. So we have,

$$(kA)^T(kA)=k^2(1+4+9)=k^2(14)=28$$

So $k=\pm \sqrt{2}$