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    <blockquote> <p>this is with multiplications of the matrices that we create our transformations</p> </blockquote> <p><em>This</em> is why we want square matrices.</p> <p>Suppose we did what you propose, and used 2x3 matrices for our transformations.</p> <p>Then a rotation would be</p> <pre><code>( x1, x2, 0 ) ( y1, y2, 0 ) </code></pre> <p>and a translation would be</p> <pre><code>( 1, 0, tx ) ( 0, 1, ty ) </code></pre> <p>and we could perform either rotations or translations by multiplying our matrix by a column vector representing the point:</p> <pre><code> ( x ) M ( y ) ( 0 ) </code></pre> <p>to get correct answers.</p> <p><strong>However</strong> - how would we go about <strong>composing</strong> transformations? Indeed, for your "for a rotation + translation I have a matrix like this" example, how did you <em>get to</em> that matrix? Sure, in this case you can just write it out, but in general? Well, you know the answer:</p> <blockquote> <p>this is with multiplications of the matrices that we create our transformations</p> </blockquote> <p>So it must be possible to <em>multiply two transformation matrices to give another transformation matrix</em>. And the rules of matrix multiplication show that this:</p> <pre><code>( . . . ) ( . . . ) ( . . . ) ( . . . ) = ??? </code></pre> <p>is not a valid matrix multiplcation. We need matrices that can be multipled in order for our transformations to be composable. So we have that extra row.</p> <hr> <p>Now, the way I've expressed it here is in fact completely backward from the standard mathematical presentation, in which the familiar transformations of rotation and translation are just special cases of the full power of homogeneous coordinate transformations on the projective plane - but I think it will do to show you why we need that extra row - to make the matrix square, and thus able to be multipled with like matrices.</p>
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    1. POWhy do 2D transformations need 3x3 matrices?
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    1. USAakashM
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    1. POWhy do 2D transformations need 3x3 matrices?
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    1. COUuh that is the point I was not seeing yet! Thanks you AakashM. Thank to have taken the time to spot the hole in my thinking.
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    2. CO+1 great answer.
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