Question

If ${{\text{A}}^2} - {\text{A}} + {\text{I}} = 0$, then the inverse of ${\text{A}}$ is:
A) ${\text{A - I}}$
B) ${\text{I - A}}$
C) ${\text{A + I}}$
D) ${\text{A}}$

Answer 1

We will solve this problem by simplifying the equation by doing multiplication with the inverse matrix. Also, we should know that the Inverse of any matrix ${\text{A}}$ is denoted by ${{\text{A}}^{ - 1}}$ , and the multiplication of that matrix and inverse of it will be equals to the identity matrix ${\text{I}}$ as shown below:
${{\text{A}}^{ - 1}}.{\text{A}} = {\text{I}}$

Complete step-by-step solution:
Step 1: It is given that ${{\text{A}}^2} - {\text{A}} + {\text{I}} = 0$ , and we need to find the inverse of the ${\text{A}}$. We will simplify the equation by multiplying ${{\text{A}}^{ - 1}}$ on both sides of the equation as below:
$\Rightarrow {{\text{A}}^{ - 1}}\left( {{{\text{A}}^2} - {\text{A}} + {\text{I}}} \right) = \left( 0 \right).{{\text{A}}^{ - 1}}$
By opening the brackets on the LHS side of the equation and multiplying the terms with ${{\text{A}}^{ - 1}}$ as shown below:
$\Rightarrow {{\text{A}}^{ - 1}}{{\text{A}}^2} - {{\text{A}}^{ - 1}}{\text{A + }}{{\text{A}}^{ - 1}}{\text{I}} = 0.{{\text{A}}^{ - 1}}$ ………………. (1)
Step 2: By replacing the terms ${{\text{A}}^{ - 1}}.{\text{A}} = {\text{I}}$ in the above equation (1), we get:
$\Rightarrow {{\text{A}}^{ - 1}}{{\text{A}}^2} - {\text{I + }}{{\text{A}}^{ - 1}}{\text{I}} = 0.{{\text{A}}^{ - 1}}$
We know that ${{\text{A}}^{ - 1}}{{\text{A}}^2} = {\text{A}}$ and ${{\text{A}}^{ - 1}}.{\text{I = }}{{\text{A}}^{ - 1}}$ , so by replacing it in the above equation ${{\text{A}}^{ - 1}}{{\text{A}}^2} - {\text{I + }}{{\text{A}}^{ - 1}}{\text{I}} = 0.{{\text{A}}^{ - 1}}$ we get:
$\Rightarrow {\text{A}} - {\text{I + }}{{\text{A}}^{ - 1}} = 0$
By bringing ${\text{A}} - {\text{I}}$ into the RHS side of the equation ${\text{A}} - {\text{I + }}{{\text{A}}^{ - 1}} = 0$, we get:
$\Rightarrow {{\text{A}}^{ - 1}} = {\text{I}} - {\text{A}}$
$\therefore$ The inverse ${\text{A}}$ will be equal to ${\text{I}} - {\text{A}}$.

Option B is the correct answer.

Note: Students need to remember that the identity matrix is a matrix that is having ones on the main diagonal and zeros elsewhere. It is denoted by a symbol ${\text{I}}$.
You should also learn some important properties of the inverse of the matrix which plays an important role in solving these types of questions. Some of the properties are mentioned below:
If ${\text{A}}$is any nonsingular matrix then the inverse of ${\text{A}}$ is ${{\text{A}}^{ - 1}}$ and
${\left( {{{\text{A}}^{ - 1}}} \right)^{ - 1}} = {\text{A}}$
If ${\text{A}}$and ${\text{B}}$are two nonsingular matrices then the product of these two matrices will also be non-singular and:
${\left( {{\text{AB}}} \right)^{ - 1}} = {{\text{A}}^{ - 1}}{{\text{B}}^{ - 1}}$
The product of a matrix and its inverse is equal to an identity matrix denoted as ${\text{I}}$:
${{\text{A}}^{ - 1}}.{\text{A}} = {\text{I}}$