= Solution
Multiplying a three-dimensional <matrix> by a real $\lambda$ multiplies its <determinant> by $\lambda^3$, whereas multiplicativity of the <determinant> gives $\det(A^5)=(\det A)^5$. Therefore equality of the two <parallelepiped> volumes means
$$
23|\lambda|^3=23^5,\qquad |\lambda|^3=23^4.
$$
Both real signs are allowed:
$$
\boxed{\lambda=\pm23^{4/3}.}
$$
A negative value reverses the <oriented volume> while preserving the ordinary volume. If a positive scale factor is intended, select the positive sign.
Back to article page
