JAMIA 2023 — Mathematics PYQ

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Question

JAMIA 2023 — Mathematics PYQ

JAMIA | Mathematics | 2023

If 𝐴2−𝐴=3𝐼 then 𝐴−1 is

Choose the correct answer:

Explanation

1. Given Equation

The equation provided is:

A^2 - A = 3I

2. Isolate the Identity Matrix

Since we want to find $A^{-1}$ , we need to manipulate the equation so that $I$ is on one side. The equation is already in that form:

A^2 - A = 3I

3. Factor out Matrix $A$

On the left side of the equation, we can factor out $A$ from both terms. Note that when we factor $A$ out of $A$ , it leaves the Identity matrix $I$ :

A(A - I) = 3I

4. Solve for $A^{-1}$

To get $A^{-1}$ , we multiply both sides of the equation by $A^{-1}$ from the left:

A^{-1} \cdot [A(A - I)] = A^{-1} \cdot (3I)

Using the property $A^{-1}A = I$ :

I(A - I) = 3A^{-1}

A - I = 3A^{-1}

5. Final Result

Now, divide both sides by 3 to isolate $A^{-1}$ :

A^{-1} = \frac{1}{3}(A - I)