Question-1

Let \(A\) be a \(3\times3\) matrix with non-zero determinant. If \(\text{det}(2A)=k\text{det}(A)\), find the value of \(k\).

---

If a row is scaled by a constant, the determinant is scaled by the same constant. When a matrix is scaled by a constant, each row of the determinant is scaled by the same constant. With this, for a \(n\times n\) matrix \(A\):

\[ \begin{aligned} \text{det}(cA) & =c^{n}\text{det}(A) \end{aligned} \]

In this question \(n=3\) and \(c=2\):

\[ \text{det}(2A)=8\text{\ensuremath{\cdot}det}(A) \]

Therefore, \(k=8\).