GATE 2024 DA Q.49: solution and answer

GATE 2024 · DAQ.49 · 2 marks · MSQ

Let $R$ be the set of real numbers, $U$ be a subspace of $R^{3}$ and $M \in R^{3 \times 3}$ be the matrix corresponding to the projection on to the subspace $U$ .

Which of the following statements is/are TRUE?

(A)
If $U$ is a 1-dimensional subspace of $R^{3}$ , then the null space of $M$ is a 1-dimensional subspace.
(B)
If $U$ is a 2-dimensional subspace of $R^{3}$ , then the null space of $M$ is a 1-dimensional subspace.
(C)
$M^{2} = M$
(D)
$M^{3} = M$

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Answer and solution

ANSWER

The true statements are (B), (C) and (D).

Official answer: B, C, DSource: Official answer key, GATE 2024 DA (IISc Bengaluru)

A projection onto $U$ sends every vector $x$ to a vector $Mx \in U$ , and leaves every vector of $U$ unchanged. So the column space of $M$ is exactly $U$ , and $rank (M) = dim U$ .
By rank-nullity for the $3 \times 3$ matrix $M$ : $dim N (M) = 3 - rank (M) = 3 - dim U$ .
(A) If $dim U = 1$ , then $dim N (M) = 3 - 1 = 2$ , not 1. So (A) is false.
(B) If $dim U = 2$ , then $dim N (M) = 3 - 2 = 1$ . So (B) is true.
(C) For any $x$ , the vector $Mx$ lies in $U$ , and $M$ fixes every vector of $U$ . Hence $M (Mx) = Mx$ for all $x$ , that is $M^{2} = M$ . So (C) is true.
(D) Using (C): $M^{3} = M M^{2} = M M = M^{2} = M$ . So (D) is true.
The true statements are B, C and D.

KEY IDEA

A projection matrix is idempotent ( $M^{2} = M$ ), its rank equals $dim U$ , and rank-nullity gives $dim N (M) = 3 - dim U$ .

WATCH OUT FOR

Confusing the dimension of the subspace being projected onto with the dimension of the null space. For a 1-dimensional $U$ the null space has dimension 2, not 1.

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GATE 2024 DA Q.49: solution and answer

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