 # 2.6 Identity Matrix

2.6 Identity Matrix
1. Identity matrix is a square matrix, usually denoted by the letter and is also known as unit matrix.

2. All the diagonal elements (from top left to bottom right) of an identity matrix are 1 and the rest of the elements are 0.
For example,

3. If is the identity matrix of order n × n and is a matrix of the same order, then IA = A and AI = A

Example 1:
Determine whether each of the following is an identity matrix of $\left(\begin{array}{cc}-2& 4\\ 3& 7\end{array}\right).$
$\left(a\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{}\left(b\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

Solution:
$\begin{array}{l}\left(a\right)\left(\begin{array}{cc}-2& 4\\ 3& 7\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\\ =\left(\begin{array}{cc}-2×1+4×0& -2×0+4×1\\ 3×1+7×0& 3×0+7×1\end{array}\right)\\ =\left(\begin{array}{cc}-2& 4\\ 3& 7\end{array}\right)\\ \text{Therefore,}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{is an identity matrix}\text{.}\\ \\ \\ \left(b\right)\left(\begin{array}{cc}-2& 4\\ 3& 7\end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\\ =\left(\begin{array}{cc}-2×0+4×1& -2×1+4×0\\ 3×0+7×1& 3×1+7×0\end{array}\right)\\ =\left(\begin{array}{cc}4& -2\\ 7& 3\end{array}\right)\\ \ne \left(\begin{array}{cc}-2& 4\\ 3& 7\end{array}\right)\\ \text{Therefore,}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\text{is not an identity matrix}\text{.}\end{array}$

Example 2:
Find the product of the following pairs of matrices and determine whether the given matrix is an identity matrix.

$\begin{array}{l}\left(a\right)\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{and}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right)\\ \left(b\right)\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)\left(\begin{array}{cc}1& 8\\ 5& 3\end{array}\right)\text{and}\left(\begin{array}{cc}1& 8\\ 5& 3\end{array}\right)\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)\end{array}$

Solution:
$\begin{array}{l}\left(a\right)\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\\ =\left(\begin{array}{cc}-3×1+2×0& -3×0+2×1\\ 5×1+7×0& 5×0+7×1\end{array}\right)=\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right)\\ \\ \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right)\\ =\left(\begin{array}{cc}1×-3+0×5& 1×2+0×7\\ 0×-3+1×5& 0×2+1×7\end{array}\right)=\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right)\\ \\ \therefore \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{is an identity matrix for}\left(\begin{array}{cc}-3& 2\\ 5& 7\end{array}\right).\\ \\ \left(b\right)\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)\left(\begin{array}{cc}1& 8\\ 5& 3\end{array}\right)\\ =\left(\begin{array}{cc}0×1+0×5& 0×8+0×3\\ 1×1+1×5& 1×8+1×3\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 6& 11\end{array}\right)\\ \\ \left(\begin{array}{cc}1& 8\\ 5& 3\end{array}\right)\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)\\ =\left(\begin{array}{cc}1×0+8×1& 1×0+8×1\\ 5×0+3×1& 5×0+3×1\end{array}\right)=\left(\begin{array}{cc}8& 8\\ 3& 3\end{array}\right)\\ \\ \therefore \left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)\text{is NOT an identity matrix for}\left(\begin{array}{cc}1& 8\\ 5& 3\end{array}\right).\end{array}$