GATE 2024 EC Q.30: solution and answer

GATE 2024 · ECQ.30 · 1 mark · NAT

Let $R$ and $R^{3}$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $α$ for which the set of vectors

{[2 - 3 α], [3 - 13], [1 - 57]}

does not form a basis of $R^{3}$ is ______.

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Answer and solution

ANSWER

The set fails to be a basis of $R^{3}$ only when $α = 5$ . At that value the determinant $70 - 14 α$ is zero, and $[2 - 35]$ is the average of the other two vectors.

Official answer: 5Source: Official answer key, GATE 2024 EC (IISc Bengaluru)

In $R^{3}$ , three vectors form a basis exactly when they are linearly independent. So the set fails to be a basis exactly when the matrix with these vectors as its rows is singular, i.e. $det = 0$ .
Put the vectors in as rows: $A = 231 - 3 - 1 - 5 α 37$
Expand the determinant along the first row (cofactor expansion): $det A = 2 [(- 1) (7) - (3) (- 5)] - (- 3) [(3) (7) - (3) (1)] + α [(3) (- 5) - (- 1) (1)]$
Simplify each minor: $(- 7 + 15) = 8$ , $(21 - 3) = 18$ , $(- 15 + 1) = - 14$ . So $det A = 2 (8) + 3 (18) + α (- 14) = 70 - 14 α$ .
For linear dependence, set $det A = 0$ : $70 - 14 α = 0 \Rightarrow α = 5$ .
Check: when $α = 5$ , $\frac{1}{2} [3 - 13] + \frac{1}{2} [1 - 57] = [2 - 35]$ . The first vector lies in the span of the other two, so the set does not form a basis of $R^{3}$ . The answer is $α = 5$ .

KEY IDEA

Three vectors in $R^{3}$ form a basis if and only if they are linearly independent, which happens if and only if the $3 \times 3$ matrix built from them has a non-zero determinant. Set the determinant to zero and solve for $α$ .

WATCH OUT FOR

Getting the sign wrong on the middle cofactor. The first-row pattern is $+, -, +$ and the entry there is $- 3$ , so the term is $- (- 3) \cdot 18 = + 54$ , not $- 54$ . With the wrong sign you get $det A = - 38 - 14 α$ and a non-integer $α$ . Also, solving for the $α$ that makes the set a basis gives every $α \neq = 5$ , not one number: the question asks for the value where it is not a basis.

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GATE 2024 EC Q.30: solution and answer

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