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# Identity Matrix

An identity matrix may be defined as any square matrix with 1s on the main diagonal and 0s everywhere else. Here are the 2 x 2 and 3 x 3 identity matrices for reference:

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

## Why is it called an identity matrix?

Identity matrices get their name from multiplication. When you multiply an identity matrix by a compatible matrix, the product is the matrix you started with. It's like the number 1 being called the multiplicative identity of real numbers because $x\left(1\right)=x$ . For instance:

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]·\left[\begin{array}{cc}\hfill 4& 7\\ \hfill -1& 3\end{array}\right]=\left[\begin{array}{cc}\hfill 1\left(4\right)+0\left(-1\right)& 1\left(7\right)+0\left(3\right)\\ \hfill 0\left(4\right)+1\left(-1\right)& 0\left(7\right)+1\left(3\right)\end{array}\right]$

$=\left[\begin{array}{cc}\hfill 4& 7\\ \hfill -1& 3\end{array}\right]$

Remember that identity matrices only work for compatible matrices. In fact, we cannot multiply matrices at all unless the first one has the same number of columns as the second one has rows. Therefore, we cannot assume that any identity matrix will act as an identity matrix in all situations.

Fun fact: all identity matrices are compatible with themselves. This makes identity matrices idempotent matrices because they satisfy the equation $A={A}^{2}$ : an identity matrix multiplied by itself will still be the identity matrix.

## Identity matrix example

a. Using the guidelines above, what would the $4×4$ identity matrix look like?

A $4×4$ matrix will have 4 rows and 4 columns, and an identity matrix has 1s across the main diagonal and 0s everywhere else. Therefore, the $4×4$ identity matrix would look like this:

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

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