Let $A$ be a symmetric matrix such that $|A| = 2$ and $\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$. If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{\alpha^2}$ is equal to ______.

### Analyzing the Setup We are given a symmetric matrix $A$. By definition, a symmetric matrix satisfies $A = A^T$. For a $2 \times 2$ matrix, this implies the off-diagonal elements are equal. We define the matrix as: $$A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$$ We are also given the determinant condition $|A| = 2$. This provides our first fundamental constraint: $$ac - b^2 = 2$$ ### The Matrix Equation We are presented with the following matrix multiplication: $$\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} \begin{bmatrix} a & b \\ b & c \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$$ Observe the structure of the multiplier matrix. The second row is exactly $\frac{3}{2}$ times the first row. Consequently, the resulting matrix must follow the same linear dependency. This allows us to determine $\alpha$ and $\beta$ immediately: $$\alpha = \frac{3}{2} \cdot 1 = \frac{3}{2}$$ $$\beta = \frac{3}{2} \cdot 2 = 3$$ ### Solving for Matrix Elements Performing the matrix multiplication explicitly, the first row of the product is $[2a+b, 2b+c]$. Equating this to the first row of the result $[1, 2]$, we obtain: $$2a + b = 1 \Rightarrow b = 1 - 2a$$ $$2b + c = 2 \Rightarrow c = 2 - 2b$$ Substituting $b$ into the expression for $c$: $$c = 2 - 2(1 - 2a) = 2 - 2 + 4a = 4a$$ Now, substitute $b = 1 - 2a$ and $c = 4a$ into the determinant equation $ac - b^2 = 2$: $$a(4a) - (1 - 2a)^2 = 2$$ $$4a^2 - (1 - 4a + 4a^2) = 2$$ The $4a^2$ terms cancel out, simplifying the equation to: $$4a - 1 = 2 \Rightarrow 4a = 3 \Rightarrow a = \frac{3}{4}$$ ### Final Calculation With $a = \frac{3}{4}$, we find $c = 4a = 3$. The trace $s$ of matrix $A$ is the sum of its diagonal elements: $$s = a + c = \frac{3}{4} + 3 = \frac{15}{4}$$ We are tasked with finding the value of $\frac{\beta s}{\alpha^2}$. Substituting our known values $\beta = 3$, $s = \frac{15}{4}$, and $\alpha = \frac{3}{2}$: $$\frac{\beta s}{\alpha^2} = \frac{3 \cdot \left( \frac{15}{4} \right)}{\left( \frac{3}{2} \right)^2} = \frac{\frac{45}{4}}{\frac{9}{4}} = \frac{45}{9} = 5$$ The final answer is **5**.

JEE Main 2023 (29 January Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Matrices and Determinants: 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 ______.

Q: Why did you assume $A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$? Is it always that simple?

Yes, the definition of a symmetric matrix is $A_{ij} = A_{ji}$. For a $2 \times 2$ matrix, this forces the off-diagonal elements to be identical. It is not an assumption; it is a geometric constraint that simplifies the problem significantly. Whenever you see the word 'symmetric' in a matrix problem, immediately write down this template to reduce your unknowns.

Enter Numerical Value:

Progressive Hints

Hint 1

Start by embracing the definition of a symmetric matrix. If

A

is symmetric, then

A = A^{T}

. This means your matrix

A

is not a collection of four independent variables, but rather a structure defined by only three:

A = [a b b c]

. This reduction in variables is your first tactical advantage in simplifying the problem.

Hint 2

Do not rush into brute-force multiplication immediately. Look closely at the multiplier matrix

[23 1 \frac{3}{2}]

. Notice that the second row is exactly

\frac{3}{2}

times the first row. This implies that the resulting matrix on the right side must also have a second row that is

\frac{3}{2}

times the first row. This observation will save you from solving a massive system of equations and reveals the hidden structure of the problem.

Hint 3

Use the determinant condition

∣ A ∣ = a c - b^{2} = 2

as your anchor. Once you express

b

and

c

in terms of

a

using the matrix multiplication results, substitute them into this determinant equation. The quadratic terms will vanish, leaving you with a clean linear equation for

a

. This is the moment where the complexity collapses into a simple, solvable form.

Visualized Solution

Defining Symmetric Matrix $A$

Let $A = [a b b c]$ be a $2 \times 2$ symmetric matrix.
For a symmetric matrix, $A = A^{T}$ , which means the non-diagonal elements are equal.

Determinant Equation $∣ A ∣ = 2$

Given the determinant $∣ A ∣ = 2$ .
For our matrix, $∣ A ∣ = a c - b^{2}$ .
Therefore, $a c - b^{2} = 2$ .

Matrix Multiplication Setup

We are given the matrix equation: $[23 1 \frac{3}{2}] A = [1 α 2 β]$
Substitute our matrix $A$ into the Left Hand Side (LHS).

Executing Matrix Multiplication

LHS: $[23 1 \frac{3}{2}] [a b b c]$
First row: $[2 a + b, 2 b + c]$
Second row: $[3 a + \frac{3}{2} b, 3 b + \frac{3}{2} c]$

Finding $α$ and $β$

Equate the resulting LHS matrix to the RHS matrix $[1 α 2 β]$ .
From row 1: $2 a + b = 1$ and $2 b + c = 2$ .
From row 2: $α = 3 a + \frac{3}{2} b$ and $β = 3 b + \frac{3}{2} c$ .

Calculating Exact Values of $α$ and $β$

Notice that $α = \frac{3}{2} (2 a + b)$ . Since $2 a + b = 1$ , $α = \frac{3}{2} (1) = \frac{3}{2}$ .
Similarly, $β = \frac{3}{2} (2 b + c)$ . Since $2 b + c = 2$ , $β = \frac{3}{2} (2) = 3$ .

Expressing Variables in Terms of $a$

From $2 a + b = 1$ , we get $b = 1 - 2 a$ .
From $2 b + c = 2$ , we get $c = 2 - 2 b$ .
Substitute $b$ into $c$ : $c = 2 - 2 (1 - 2 a) = 2 - 2 + 4 a = 4 a$ .

Solving for $a$

Recall the determinant equation: $a c - b^{2} = 2$ .
Substitute $b = 1 - 2 a$ and $c = 4 a$ : $a (4 a) - (1 - 2 a)^{2} = 2$ .
Expand: $4 a^{2} - (1 - 4 a + 4 a^{2}) = 2$ .
Simplify: $4 a^{2} - 1 + 4 a - 4 a^{2} = 2 ⟹ 4 a - 1 = 2$ .
Solve: $4 a = 3 ⟹ a = \frac{3}{4}$ .

Calculating Trace $s$

We know $c = 4 a$ . Substitute $a = \frac{3}{4}$ : $c = 4 (\frac{3}{4}) = 3$ .
The sum of diagonal elements is $s = a + c$ .
$s = \frac{3}{4} + 3 = \frac{3}{4} + \frac{12}{4} = \frac{15}{4}$ .

Final Evaluation of $\frac{β s}{α ^{2}}$

We need to find the value of $\frac{β s}{α ^{2}}$ .
Substitute known values: $α = \frac{3}{2}$ , $β = 3$ , $s = \frac{15}{4}$ .
$\frac{β s}{α ^{2}} = \frac{3 \cdot ( \frac{15}{4} )}{( \frac{3}{2} ) ^{2}} = \frac{\frac{45}{4}}{\frac{9}{4}}$ .
Simplify: $\frac{45}{4} \cdot \frac{4}{9} = \frac{45}{9} = 5$ .

00:00 / 00:00

The Sigma Insight: Types of Matrices

Analyzing the Setup

We are given a symmetric matrix

A

. By definition, a symmetric matrix satisfies

A = A^{T}

. For a

2 \times 2

matrix, this implies the off-diagonal elements are equal.

We define the matrix as:

A = [a b b c]

We are also given the determinant condition

∣ A ∣ = 2

. This provides our first fundamental constraint:

a c - b^{2} = 2

The Matrix Equation

We are presented with the following matrix multiplication:

[23 1 \frac{3}{2}] [a b b c] = [1 α 2 β]

Observe the structure of the multiplier matrix. The second row is exactly

\frac{3}{2}

times the first row. Consequently, the resulting matrix must follow the same linear dependency.

This allows us to determine

α

and

β

immediately:

α = \frac{3}{2} \cdot 1 = \frac{3}{2}

β = \frac{3}{2} \cdot 2 = 3

Solving for Matrix Elements

Performing the matrix multiplication explicitly, the first row of the product is

[2 a + b, 2 b + c]

. Equating this to the first row of the result

[1, 2]

, we obtain:

2 a + b = 1 \Rightarrow b = 1 - 2 a

2 b + c = 2 \Rightarrow c = 2 - 2 b

Substituting

b

into the expression for

c

c = 2 - 2 (1 - 2 a) = 2 - 2 + 4 a = 4 a

Now, substitute

b = 1 - 2 a

and

c = 4 a

into the determinant equation

a c - b^{2} = 2

a (4 a) - (1 - 2 a)^{2} = 2

4 a^{2} - (1 - 4 a + 4 a^{2}) = 2

The

4 a^{2}

terms cancel out, simplifying the equation to:

4 a - 1 = 2 \Rightarrow 4 a = 3 \Rightarrow a = \frac{3}{4}

Final Calculation

With

a = \frac{3}{4}

, we find

c = 4 a = 3

. The trace

s

of matrix

A

is the sum of its diagonal elements:

s = a + c = \frac{3}{4} + 3 = \frac{15}{4}

We are tasked with finding the value of

\frac{β s}{α ^{2}}

. Substituting our known values

β = 3

s = \frac{15}{4}

, and

α = \frac{3}{2}

\frac{β s}{α ^{2}} = \frac{3 \cdot ( \frac{15}{4} )}{( \frac{3}{2} ) ^{2}} = \frac{\frac{45}{4}}{\frac{9}{4}} = \frac{45}{9} = 5

The final answer is 5.

Common Traps & FAQs

Why did you assume

A = [a b b c]

? Is it always that simple?

Yes, the definition of a symmetric matrix is

A_{ij} = A_{j i}

. For a

2 \times 2

matrix, this forces the off-diagonal elements to be identical. It is not an assumption; it is a geometric constraint that simplifies the problem significantly. Whenever you see the word 'symmetric' in a matrix problem, immediately write down this template to reduce your unknowns.

Is there a faster way than multiplying the whole matrix? It looks tedious.

Absolutely! As hinted, notice the row dependency in the multiplier matrix. If you see that the second row is a scalar multiple of the first, you can deduce the relationship between

α

and

β

without full expansion. However, for the sake of clarity and to avoid silly mistakes, full multiplication is a safe and robust path that confirms your intuition.

Why did the

4 a^{2}

terms cancel out? Did I make a mistake?

That is the beauty of a well-constructed JEE problem! When you substitute

b = 1 - 2 a

and

c = 4 a

into

a c - b^{2} = 2

, the quadratic terms

4 a^{2}

and

- (2 a)^{2}

cancel perfectly. If they did not, it would suggest a calculation error. Trust the algebra and enjoy the moment when the complexity disappears!

Animated Solution for Mathematics - Matrices and Determinants: 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 ______.

Enter Numerical Value:

Visualized Solution

Defining Symmetric Matrix $A$

Determinant Equation $∣ A ∣ = 2$

Matrix Multiplication Setup

Executing Matrix Multiplication

Finding $α$ and $β$

Calculating Exact Values of $α$ and $β$

Expressing Variables in Terms of $a$

Solving for $a$

Calculating Trace $s$

Final Evaluation of $\frac{β s}{α ^{2}}$

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Matrix Equation

Solving for Matrix Elements

Final Calculation

Similar Questions

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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 $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Enter Numerical Value:

Visualized Solution

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

Determinant Equation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣A∣=2

Matrix Multiplication Setup

Executing Matrix Multiplication

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β

Calculating Exact Values of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β

Expressing Variables in Terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

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

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

Final Evaluation of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α2βs​

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Matrix Equation

Solving for Matrix Elements

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 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; } The number of 3×2 matrices A, which can be formed using the elements of the set {−2,−1,0,1,2} such that the sum of all the diagonal elements of ATA is 5, is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A is a symmetric matrix and B is a skew-symmetric matrix such that A+B=(25​3−1​), then AB is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The total number of 3×3 matrices A having entries from the set {0,1,2,3} such that the sum of all the diagonal entries of AAT is 9, 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 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 and B be any two 3×3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Defining Symmetric Matrix $A$

Determinant Equation $∣ A ∣ = 2$

Finding $α$ and $β$

Calculating Exact Values of $α$ and $β$

Expressing Variables in Terms of $a$

Solving for $a$

Calculating Trace $s$

Final Evaluation of $\frac{β s}{α ^{2}}$

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:

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If A is a symmetric matrix and B is a skew-symmetric matrix such that $A + B = (25 3 - 1)$ , then AB is equal to :

The total number of $3 \times 3$ matrices $A$ having entries from the set ${0, 1, 2, 3}$ such that the sum of all the diagonal entries of $A A^{T}$ is 9, is equal to ______

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 $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?