GATE ECE 2024 | Question: 20

Question

Let $\mathbb{R}$ and $\mathbb{R}^{3}$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors
\[
\left\{\left[\begin{array}{lll}
2 & -3 & \alpha
\end{array}\right], \quad\left[\begin{array}{lll}
3 & -1 & 3
\end{array}\right], \quad\left[\begin{array}{lll}
1 & -5 & 7
\end{array}\right]\right\}
\]
does not form a basis of $\mathbb{R}^{3}$ is $\_\_\_\_\_\_\_$.

Answer 1

5

Answer 2

2	-3	a
3	-1	3
1	-5	7

2(-7+15)-3(-21+5a)+1(-9+a)=0

16+63-15a-9+a=0

70=14a

a=5