Question

What is the multiplicative inverse of a matrix?

Answer 1

Hint: In this problem we need to calculate the multiplicative inverse of a matrix. First, we will assume a matrix $A$ and the inverse of the matrix $A$ as $B$. When we have multiplied a matrix with the given matrix, if we get the identity matrix as a result then we can say that the multiplied matrix is called a multiplicative inverse of the given matrix. So, we can write the mathematical form of the statement in terms of $A$ and $B$. Now we will simplify the equation to get the value of $B$ which is our required result.

Complete step by step solution:
Let us assume a matrix which is named as $A$.
Let the multiplicative inverse of the matrix $A$ is $B$.
We know that the product of the matrix $A$ and its multiplicative inverse which is $B$ should be equal to the identity matrix. So, we can write this mathematically as
$A\times B=I$
Dividing the above equation with $A$ on both sides, then we will get
${\displaystyle \frac{AB}{A}}={\displaystyle \frac{I}{A}}$
Cancelling the term $A$ which is in both numerator and denominator on left hand side, then we will have
$B={\displaystyle \frac{I}{A}}$
We can write the term ${\displaystyle \frac{1}{A}}$ as $A^{-1}$. Substituting this value in the above equation, then we will get
$B=I\times A^{-1}$
We know that the product of any matrix with identity matrix will be equal to a multiplied matrix. So, we can write the above equation as
$B=A^{-1}$
Hence the multiplicative inverse of the matrix$\left(A\right)$ is equal to the inverse of that matrix$\left(A^{-1}\right)$.

Note: The terms inverse of the matrix and multiplicative inverse of the matrix are used in the same meaning. So, whenever someone asked to calculate the multiplicative inverse then simply calculate the inverse of the matrix from the formula $A^{-1}={\displaystyle \frac{1}{\left|A\right|}}adj\left(A\right)$.