## Precalculus (6th Edition) Blitzer

The multiplicative identity of the matrix is denoted by ${{I}_{N}}$. The $n\times n$ square matrix consists of $n$ elements on the diagonal with the value 1 and 0s elsewhere. The multiplicative identity is the property of multiplication that states: when 1 is multiplied by any real number, the real number does not change; therefore the number 1 is called the multiplicative identity for real numbers. Consider a matrix, such that, \begin{align} & {{I}_{1}}=\left[ 1 \right] \\ & {{I}_{2}}=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & {{I}_{3}}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \\ \end{align} These are the examples of identity matrices of order $1\times 1,2\times 2,3\times 3,\ldots ,n\times n.$ Consider, $M=\left[ \begin{matrix} -4 & -3 \\ -6 & 5 \\ \end{matrix} \right]$ This is a square matrix of order $2\times 2$. The identity matrix is, $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$ Now, the rule of matrix multiplication is $A=MI$ Then, \begin{align} & A=\left[ \begin{matrix} -4 & -3 \\ -6 & 5 \\ \end{matrix} \right]\times \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -4 & -3 \\ -6 & 5 \\ \end{matrix} \right] \end{align} Hence, the identity matrix is $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$.