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When multiplying numbers, the number 1 has a special property: when you multiply 1 by any number, you get that same number back. We can express this property as an algebraic generalization:
The matrix that has this property is referred to as the identity matrix .
Note that the definition of [I] stipulates that the multiplication must commute —that is, it must yield the same answer no matter which order you multiply in. This is important because, for most matrices, multiplication does not commute.
What matrix has this property? Your first guess might be a matrix full of 1s, but that doesn’t work:
$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ = $\left[\begin{array}{cc}3& 3\\ 7& 7\end{array}\right]$ | so $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ is not an identity matrix |
The matrix that does work is a diagonal stretch of 1s, with all other elements being 0.
$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ = $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ | so $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ is the identity for 2x2 matrices |
$\left[\begin{array}{ccc}2& 5& 9\\ \pi & -2& 8\\ -3& 1/2& 8\text{.}3\end{array}\right]$ $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ = $\left[\begin{array}{ccc}2& 5& 9\\ \pi & -2& 8\\ -3& 1/2& 8\text{.}3\end{array}\right]$ | $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ is the identity for 3x3 matrices |
You should confirm those multiplications for yourself, and also confirm that they work in reverse order (as the definition requires).
Hence, we are led from the definition to:
Why no identity for a non-square matrix? Because of the requirement of commutativity. For a non-square matrix $\left[A\right]$ you might be able to find a matrix $\left[I\right]$ such that $\left[A\right]\left[I\right]=\left[A\right]$ ; however, if you reverse the order, you will be left with an illegal multiplication.
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