Chapter 8 - Section 8.4 - Multiplicative Inverses of Matrices and Matrix Equations - Concept and Vocabulary Check - Page 931: 3

For $n\times n$ matrix $A$ and $B$, if $AB={{I}_{n}}$ and $BA={{I}_{n}}$, then $B$ is called the Multiplicative Inverse of $A$.

The multiplicative identity of the matrix is denoted by ${{I}_{N}}$. In the $n\times n$ square matrix, we have $n$ elements with 1 down the main diagonal and 0s elsewhere. The multiplicative identity is the property of multiplication which states that when 1 is multiplied by any real number, the real number does not change; the number 1 is called the multiplicative identity for real numbers. Consider the identity matrices: $\begin{align} & {{I}_{1}}=\left[ 1 \right] \\ & {{I}_{2}}=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & {{I}_{3}}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \\ \end{align}$ These are the examples of identity matrices of order $1\times 1,2\times 2,3\times 3...........,n\times n.$ As example: let $A=\left[ \begin{matrix} 4 & -3 \\ -5 & 4 \\ \end{matrix} \right],B=\left[ \begin{matrix} 4 & 3 \\ 5 & 4 \\ \end{matrix} \right]$ Now, we will compute the matrix as $\left[ AB \right]$ $\begin{align} & AB=\left[ \begin{matrix} 4 & -3 \\ -5 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} 4 & 3 \\ 5 & 4 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4\times 4+\left( -3 \right)\times 5 & 4\times 3+\left( -3 \right)\times 4 \\ \left( -5 \right)\times 4+4\times 5 & \left( -5 \right)\times 3+4\times 4 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & ={{I}_{2}} \end{align}$ Now we will compute the matrix as $\left[ BA \right]$ $\begin{align} & BA=\left[ \begin{matrix} 4 & 3 \\ 5 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} 4 & -3 \\ -5 & 4 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4\times 4+3\times \left( -5 \right) & 4\times \left( -3 \right)+3\times 4 \\ 5\times 4+4\times \left( -5 \right) & 5\times \left( -3 \right)+4\times 4 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & ={{I}_{2}} \end{align}$ Thus, both matrices $AB$ and $BA$ are equal to the identity matrix.