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The projection of a vector on a plane is its orthogonal projection on that plane.

A square matrix P is called an orthogonal projection if it is both idempotent and symmetric.

i.e. $P^2=P\mathrm{and}{P}^T$

where${\mathrm{P}}^{\mathrm{T}}$ is the transpose of the matrix P.

If P is the projection matrix, i.e. ${\mathrm{P}}^2=\mathrm{P}\mathrm{and}{\mathrm{P}}^{\mathrm{T}}$then the matrix P is similar to the matrix of the type $\left[\begin{array}{cc}{I}_m& 0\\ 0& 0\end{array}\right]$, where$I_m$ is the identity matrix of order 鈥榤鈥�.

And the rank of the projection matrix is equal to the trace of the projection matrix.

Since it is given thatA is a 3 x 3 matrix that represents the projection onto a plane in${\mathrm{鈩�}}^3$,therefore, the matrix A is similar to$\left[\begin{array}{cc}{I}_m& 0\\ 0& 0\end{array}\right]$ where m=2.

$鈬�A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Sincethe rank of the projection matrix is equal to the trace of the projection matrix, then rank(A)=trace(A).

Now, trace of A = 1+1+0 = 2

$鈬�\mathrm{rank}\left(\mathrm{A}\right)=\mathrm{trace}\left(\mathrm{A}\right)=2$

If a 3 x 3 matrix A represents the projection onto a plane in ${\mathrm{鈩�}}^3$, then rank(A)=2.