JEE Main 2024 (01 Feb Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Matrices and Determinants: Let $A = I_{2} - 2 M M^{T}$ , where $M$ is real matrix of order $2 \times 1$ such that the relation $M^{T} M = I_{1}$ holds. If $λ$ is a real number such that the relation $A X = λ X$ holds for some non-zero real matrix $X$ of order $2 \times 1$ , then the sum of squares of all possible values of $λ$ is equal to :

Q: Wait, why is $M^TM = 1$? I thought $M^TM$ was a matrix?

Great question! $M$ is a $2 \times 1$ matrix, so $M^T$ is a $1 \times 2$ matrix. The product $M^TM$ is $(1 \times 2) \times (2 \times 1)$, which results in a $1 \times 1$ matrix. In linear algebra, a $1 \times 1$ matrix is effectively a scalar. So, $M^TM = [1]$ is equivalent to the scalar $1$.

Q: Why can we just say $1 - \lambda^2 = 0$ when $(1 - \lambda^2)X = 0$?

This is a fundamental property of linear systems. If $cX = 0$ and $X$ is a non-zero vector, then the scalar $c$ must be zero. If $c$ were non-zero, we could multiply by $1/c$ to get $X = 0$, which contradicts our premise that $X$ is non-zero. Thus, the scalar coefficient must vanish.

Enter Numerical Value:

Visualized Solution

Understanding Matrix $A$

Given matrix $A = I_{2} - 2 M M^{T}$
Matrix $M$ is of order $2 \times 1$ (Column Matrix)
$M^{T}$ is of order $1 \times 2$ (Row Matrix)
Therefore, $M M^{T}$ is a $2 \times 2$ matrix

The Property of Matrix $M$

Given constraint: $M^{T} M = I_{1}$
Since $I_{1}$ is a $1 \times 1$ identity matrix, we can treat $M^{T} M = 1$

Squaring Matrix $A$

To find the properties of $A$ , let's calculate $A^{2}$ :
$A^{2} = (I_{2} - 2 M M^{T}) (I_{2} - 2 M M^{T})$

Expanding the Product

Expanding using distributive property:
$A^{2} = I_{2} \cdot I_{2} - I_{2} (2 M M^{T}) - (2 M M^{T}) I_{2} + (2 M M^{T}) (2 M M^{T})$
$A^{2} = I_{2} - 4 M M^{T} + 4 (M M^{T}) (M M^{T})$

Using Associativity

Focusing on the term $(M M^{T}) (M M^{T})$ :
By matrix associativity: $M (M^{T} M) M^{T}$
Substitute $M^{T} M = 1$ :
$M (1) M^{T} = M M^{T}$

Simplifying $A^{2}$

Substitute back into the $A^{2}$ expression:
$A^{2} = I_{2} - 4 M M^{T} + 4 (M M^{T})$
$A^{2} = I_{2}$

The Eigenvalue Relation

Given relation: $A X = λ X$
Pre-multiply by $A$ on both sides:
$A (A X) = A (λ X)$

Relating to $A^{2}$

$A^{2} X = λ (A X)$
Substitute $A X = λ X$ and $A^{2} = I_{2}$ :
$I_{2} X = λ (λ X)$
$X = λ^{2} X$

Solving for $λ$

Rearranging the equation:
$X - λ^{2} X = 0$
$(1 - λ^{2}) X = 0$
Since $X \neq = 0$ , we must have $1 - λ^{2} = 0$
$λ^{2} = 1 ⟹ λ = 1, - 1$

Final Calculation

Possible values of $λ$ are $1$ and $- 1$ .
We need the sum of squares of all possible values of $λ$ .
Sum of squares $= (1)^{2} + (- 1)^{2}$
Sum of squares $= 1 + 1 = 2$
Final Answer: 2

00:00 / 00:00

The Sigma Insight: Types of Matrices

Analyzing the Setup

We are given the matrix

A = I_{2} - 2 M M^{T}

, where

M

is a

2 \times 1

column matrix satisfying the constraint

M^{T} M = 1

In the realm of linear algebra, this specific form is recognized as a Householder transformation, which geometrically represents a reflection.

The Power of Squaring

To understand the nature of matrix

A

, we examine its behavior under squaring:

A^{2} = (I_{2} - 2 M M^{T}) (I_{2} - 2 M M^{T})

Expanding this expression using the distributive property, we obtain:

A^{2} = I_{2} - 2 M M^{T} - 2 M M^{T} + 4 (M M^{T}) (M M^{T})

A^{2} = I_{2} - 4 M M^{T} + 4 (M M^{T}) (M M^{T})

The Associativity Trick

We utilize the associative property of matrix multiplication to simplify the term

(M M^{T}) (M M^{T})

. We regroup the terms as follows:

(M M^{T}) (M M^{T}) = M (M^{T} M) M^{T}

Given the constraint

M^{T} M = 1

, the expression collapses:

M (1) M^{T} = M M^{T}

Substituting this back into our expansion for

A^{2}

A^{2} = I_{2} - 4 M M^{T} + 4 (M M^{T})

A^{2} = I_{2}

This confirms that

A

is an involutory matrix, meaning it is its own inverse.

The Eigenvalue Connection

We consider the eigenvalue equation

A X = λ X

. Pre-multiplying both sides by

A

, we get:

A (A X) = A (λ X)

A^{2} X = λ (A X)

Since

A^{2} = I_{2}

, the equation simplifies to:

X = λ (λ X)

X = λ^{2} X

This implies

(1 - λ^{2}) X = 0

. For a non-zero eigenvector

X

, we must have:

λ^{2} = 1 \Rightarrow λ = 1, - 1

Final Calculation

The possible eigenvalues of matrix

A

are

1

and

- 1

. The problem asks for the sum of the squares of all possible values of

λ

Sum = (1)^{2} + (- 1)^{2} = 1 + 1 = 2

The final result is 2.

Common Traps & FAQs

Wait, why is

M^{T} M = 1

? I thought

M^{T} M

was a matrix?

Great question!

M

is a

2 \times 1

matrix, so

M^{T}

is a

1 \times 2

matrix. The product

M^{T} M

(1 \times 2) \times (2 \times 1)

, which results in a

1 \times 1

matrix. In linear algebra, a

1 \times 1

matrix is effectively a scalar. So,

M^{T} M = [1]

is equivalent to the scalar

1

Why can we just say

1 - λ^{2} = 0

when

(1 - λ^{2}) X = 0

This is a fundamental property of linear systems. If

c X = 0

and

X

is a non-zero vector, then the scalar

c

must be zero. If

c

were non-zero, we could multiply by

1/ c

to get

X = 0

, which contradicts our premise that

X

is non-zero. Thus, the scalar coefficient must vanish.

Is there a geometric interpretation of

A = I_{2} - 2 M M^{T}

Yes! This is a Householder transformation. It represents a reflection across a line or plane. Reflections are involutory, meaning if you reflect twice, you end up back where you started. That's why

A^{2} = I_{2}

Enter Numerical Value:

Visualized Solution

Understanding Matrix $A$

The Property of Matrix $M$

Squaring Matrix $A$

Expanding the Product

Using Associativity

Simplifying $A^{2}$

The Eigenvalue Relation

Relating to $A^{2}$

Solving for $λ$

Final Calculation

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Power of Squaring

The Associativity Trick

The Eigenvalue Connection

Final Calculation

Similar Questions

Let $A$ be a square matrix such that $A A^{T} = I$ . Then $\frac{1}{2} A [(A + A^{T})^{2} + (A - A^{T})^{2}]$ is equal to

Let $A$ be a symmetric matrix such that $∣ A ∣ = 2$ and $[23 1 \frac{3}{2}] A = [1 α 2 β]$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac{β s}{α ^{2}}$ is equal to ______.

Let $a, b, c \in R$ be all non-zero and satisfy $a^{3} + b^{3} + c^{3} = 2$ . If the matrix $A = a b c b c a c a b$ satisfies $A A^{T} = I$ , then a value of $ab c$ can be :

If $A = \frac{1}{2} [1 - 3 31]$ , then :

Let $A$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^{2}$ is 1, then the possible number of such matrices is:

Let $A = [α 6 - 1 β]$ , $α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If $I$ denotes $2 \times 2$ identity matrix, then the matrix $(I + A)^{8}$ is:

Let $A = 0 p p 2 q q - q r - r r$ . If $A A^{T} = I_{3}$ , then $∣ p ∣$ is :

Let $A = [2 a 30], a \in R$ be written as $P + Q$ where $P$ is a symmetric matrix and $Q$ is a skew symmetric matrix. If $det (Q) = 9$ , then the modulus of the sum of all possible values of determinant of $P$ is equal to :

Enter Numerical Value:

Visualized Solution

Understanding Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

The Property of Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M

Squaring Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Expanding the Product

Using Associativity

Simplifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A2

The Eigenvalue Relation

Relating to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A2

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } λ

Final Calculation

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Power of Squaring

The Associativity Trick

The Eigenvalue Connection

Final Calculation

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a square matrix such that AAT=I. Then 21​A[(A+AT)2+(A−AT)2] is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a symmetric matrix such that ∣A∣=2 and [23​123​​]A=[1α​2β​]. If the sum of the diagonal elements of A is s, then α2βs​ is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c∈R be all non-zero and satisfy a3+b3+c3=2. If the matrix A=​abc​bca​cab​​ satisfies AAT=I, then a value of abc can be :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=21​[1−3​​3​1​], then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[α6​−1β​], α>0, such that det(A)=0 and α+β=1. If I denotes 2×2 identity matrix, then the matrix (I+A)8 is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​0pp​2qq−q​r−rr​​. If AAT=I3​, then ∣p∣ is :

Understanding Matrix $A$

The Property of Matrix $M$

Squaring Matrix $A$

Simplifying $A^{2}$

Relating to $A^{2}$

Solving for $λ$

Let $A$ be a square matrix such that $A A^{T} = I$ . Then $\frac{1}{2} A [(A + A^{T})^{2} + (A - A^{T})^{2}]$ is equal to

Let $A$ be a symmetric matrix such that $∣ A ∣ = 2$ and $[23 1 \frac{3}{2}] A = [1 α 2 β]$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac{β s}{α ^{2}}$ is equal to ______.

Let $a, b, c \in R$ be all non-zero and satisfy $a^{3} + b^{3} + c^{3} = 2$ . If the matrix $A = a b c b c a c a b$ satisfies $A A^{T} = I$ , then a value of $ab c$ can be :

If $A = \frac{1}{2} [1 - 3 31]$ , then :

Let $A$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^{2}$ is 1, then the possible number of such matrices is:

Let $A = [α 6 - 1 β]$ , $α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If $I$ denotes $2 \times 2$ identity matrix, then the matrix $(I + A)^{8}$ is:

Let $A = 0 p p 2 q q - q r - r r$ . If $A A^{T} = I_{3}$ , then $∣ p ∣$ is :