Let $A$ be a $3 \times 3$ real matrix. If $\det(2 \text{ Adj}(2 \text{ Adj}(\text{Adj}(2 A)))) = 2^{41}$, then the value of $\det(A^2)$ equal .

### Analyzing the Setup The problem asks us to solve for the determinant of a matrix $A$ of order $3 \times 3$ given the equation: $$\det(2 \text{ Adj}(2 \text{ Adj}(\text{Adj}(2 A)))) = 2^{41}$$ To dismantle this, we rely on two fundamental properties for an $n \times n$ matrix $M$ where $n=3$: 1. The scalar property: $\det(kM) = k^3 \det(M)$. 2. The adjoint property: $\det(\text{Adj } M) = (\det M)^{3-1} = (\det M)^2$. ### Peeling the Layers We begin by extracting the outermost scalar $2$ from the determinant. Since the matrix is $3 \times 3$, we have: $$\det(2 \text{ Adj}(2 \text{ Adj}(\text{Adj}(2 A)))) = 2^3 \det(\text{Adj}(2 \text{ Adj}(\text{Adj}(2 A))))$$ Applying the adjoint property $\det(\text{Adj } M) = (\det M)^2$ to the term inside, we get: $$2^3 [\det(2 \text{ Adj}(\text{Adj}(2 A)))]^2$$ Next, we extract the scalar $2$ from the interior determinant. Because it is inside a square, the factor $2^3$ becomes $(2^3)^2 = 2^6$: $$2^3 \cdot 2^6 \cdot [\det(\text{Adj}(\text{Adj}(2 A)))]^2 = 2^9 [\det(\text{Adj}(\text{Adj}(2 A)))]^2$$ ### The Final Descent We now simplify the nested adjoints. Using the property $\det(\text{Adj}(\text{Adj } M)) = (\det M)^{(n-1)^2}$, for $n=3$ we have $(\det M)^4$. Applying this to our expression: $$2^9 [(\det(2A))^4]^2 = 2^9 (\det(2A))^8$$ We extract the scalar $2$ from $\det(2A)$, which yields $2^3 \det(A)$. Raising this to the power of $8$: $$2^9 \cdot (2^3 \det A)^8 = 2^9 \cdot 2^{24} \cdot (\det A)^8 = 2^{33} (\det A)^8$$ Equating this to the given value $2^{41}$: $$2^{33} (\det A)^8 = 2^{41}$$ $$(\det A)^8 = 2^8$$ This implies $\det A = \pm 2$. Therefore, the final result is: $$\det(A^2) = (\det A)^2 = (\pm 2)^2 = \mathbf{4}$$

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Matrices and Determinants: Let $A$ be a $3 \times 3$ real matrix. If $det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41}$ , then the value of $det (A^{2})$ equal .

Enter Numerical Value:

Visualized Solution

Understanding the Core Objective

Given: $A$ is a $3 \times 3$ real matrix.
Equation: $det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41}$ .
Objective: Find $det (A^{2})$ .

The Mathematical Toolset

Property 1: For an $n \times n$ matrix $M$ , $det (k M) = k^{n} det (M)$ .
Property 2: For an $n \times n$ matrix $M$ , $det (Adj M) = (det M)^{n - 1}$ .
For $n = 3$ , $det (k M) = k^{3} det (M)$ and $det (Adj M) = (det M)^{2}$ .

Peeling the Outermost Scalar

Apply $det (k M) = k^{3} det M$ to the outermost $2$ :
$det (2 Adj (2 Adj (Adj (2 A)))) = 2^{3} det (Adj (2 Adj (Adj (2 A))))$

Handling the First Adjoint

Apply $det (Adj M) = (det M)^{2}$ :
$= 2^{3} [det (2 Adj (Adj (2 A)))]^{2}$

Extracting the Inner Scalar

Extract the inner scalar $2$ :
$= 2^{3} [2^{3} det (Adj (Adj (2 A)))]^{2}$
$= 2^{3} [2^{6} (det (Adj (Adj (2 A))))^{2}]$

Simplifying the Nested Adjoints

Using $det (Adj (Adj M)) = (det M)^{4}$ :
$= 2^{9} [(det (2 A))^{2}]^{2} = 2^{9} (det (2 A))^{4}$
Wait, let's re-evaluate carefully: $(det (Adj (Adj (2 A))))^{2} = ((det (Adj (2 A)))^{2})^{2} = (det (Adj (2 A)))^{4}$ .
Then, $(det (Adj (2 A)))^{4} = ((det (2 A))^{2})^{4} = (det (2 A))^{8}$ .

The Final Scalar and Matrix A

Substitute $det (2 A) = 2^{3} det A$ :
$= 2^{9} (2^{3} det A)^{8}$
$= 2^{9} \cdot 2^{24} (det A)^{8}$
$= 2^{33} (det A)^{8}$

Solving for the Determinant of A

Equate to the given value:
$2^{33} (det A)^{8} = 2^{41}$
$(det A)^{8} = \frac{2 ^{41}}{2 ^{33}} = 2^{8}$
$⟹ det A = \pm 2$

Finding the Final Answer

Calculate $det (A^{2})$ :
$det (A^{2}) = (det A)^{2}$
Substituting $det A = \pm 2$ :
$det (A^{2}) = (\pm 2)^{2} = 4$

00:00 / 00:00

The Sigma Insight: Adjoint and Inverse of a Matrix

Analyzing the Setup

The problem asks us to solve for the determinant of a matrix

A

of order

3 \times 3

given the equation:

det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41}

To dismantle this, we rely on two fundamental properties for an

n \times n

matrix

M

where

n = 3

1. The scalar property:

det (k M) = k^{3} det (M)

2. The adjoint property:

det (Adj M) = (det M)^{3 - 1} = (det M)^{2}

Peeling the Layers

We begin by extracting the outermost scalar

2

from the determinant. Since the matrix is

3 \times 3

, we have:

det (2 Adj (2 Adj (Adj (2 A)))) = 2^{3} det (Adj (2 Adj (Adj (2 A))))

Applying the adjoint property

det (Adj M) = (det M)^{2}

to the term inside, we get:

2^{3} [det (2 Adj (Adj (2 A)))]^{2}

Next, we extract the scalar

2

from the interior determinant. Because it is inside a square, the factor

2^{3}

becomes

(2^{3})^{2} = 2^{6}

2^{3} \cdot 2^{6} \cdot [det (Adj (Adj (2 A)))]^{2} = 2^{9} [det (Adj (Adj (2 A)))]^{2}

The Final Descent

We now simplify the nested adjoints. Using the property

det (Adj (Adj M)) = (det M)^{(n - 1)^{2}}

, for

n = 3

we have

(det M)^{4}

. Applying this to our expression:

2^{9} [(det (2 A))^{4}]^{2} = 2^{9} (det (2 A))^{8}

We extract the scalar

2

from

det (2 A)

, which yields

2^{3} det (A)

. Raising this to the power of

8

2^{9} \cdot (2^{3} det A)^{8} = 2^{9} \cdot 2^{24} \cdot (det A)^{8} = 2^{33} (det A)^{8}

Equating this to the given value

2^{41}

2^{33} (det A)^{8} = 2^{41}

(det A)^{8} = 2^{8}

This implies

det A = \pm 2

. Therefore, the final result is:

det (A^{2}) = (det A)^{2} = (\pm 2)^{2} = 4

Common Traps & FAQs

Wait, why does the scalar 2 come out as 2^3? Is it always 2^3?

Great question! The property is det(kA) = k^n * det(A), where n is the order of the matrix. Since we are dealing with a 3x3 matrix, n = 3. So, the scalar 2 must be raised to the power of 3 when it exits the determinant. If it were a 2x2 matrix, it would be 2^2. Always check the dimension of the matrix first!

Why can't we just use energy conservation or some other physics principle?

This is a pure linear algebra problem, not physics! In JEE Advanced, you will encounter problems that test your mathematical toolkit. Here, the 'energy' is the determinant property. Think of these properties as the laws of physics for matrices—they are the only tools that will unlock the solution.

Why do we get plus or minus 2 for det(A)? Does the sign matter?

When you have an equation like (det(A))^8 = 2^8, taking the eighth root (an even root) mathematically yields both positive and negative solutions: det(A) = ±2. However, the question asks for det(A^2). Since det(A^2) = (det(A))^2, squaring either +2 or -2 results in the same value: 4. The sign disappears, and the answer remains robust!

Animated Solution for Mathematics - Matrices and Determinants: Let $A$ be a $3 \times 3$ real matrix. If $det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41}$ , then the value of $det (A^{2})$ equal .

Enter Numerical Value:

Visualized Solution

Understanding the Core Objective

The Mathematical Toolset

Peeling the Outermost Scalar

Handling the First Adjoint

Extracting the Inner Scalar

Simplifying the Nested Adjoints

The Final Scalar and Matrix A

Solving for the Determinant of A

Finding the Final Answer

The Sigma Insight: Adjoint and Inverse of a Matrix

Analyzing the Setup

Peeling the Layers

The Final Descent

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a 3×3 real matrix. If det(2 Adj(2 Adj(Adj(2A))))=241, then the value of det(A2) equal .

Enter Numerical Value:

Visualized Solution

Understanding the Core Objective

The Mathematical Toolset

Peeling the Outermost Scalar

Handling the First Adjoint

Extracting the Inner Scalar

Simplifying the Nested Adjoints

The Final Scalar and Matrix A

Solving for the Determinant of A

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The Sigma Insight: Adjoint and Inverse of a Matrix

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