JEE Main 2020 (8 January Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Matrices and Determinants: The number of all $3 \times 3$ matrices A, with entries from the set ${- 1, 0, 1}$ such that the sum of the diagonal elements of $(A A^{T})$ is 3, is __________.

Enter Numerical Value:

Visualized Solution

Defining Matrix $A$

Let $A$ be a $3 \times 3$ matrix.
Entries $a_{ij} \in {- 1, 0, 1}$ .

Trace of $A A^{T}$

The sum of diagonal elements of a matrix is its Trace.
We need to analyze $t r (A A^{T})$ .

Property of $t r (A A^{T})$

Standard Property: $t r (A A^{T}) = \sum_{i = 1}^{3} \sum_{j = 1}^{3} a_{ij}^{2}$
This is the sum of squares of all $9$ elements of matrix $A$ .

Setting up the Equation

Given: $t r (A A^{T}) = 3$
$⟹ \sum a_{ij}^{2} = 3$

Analyzing Possible Values

Since $a_{ij} \in {- 1, 0, 1}$ , squaring them gives $a_{ij}^{2} \in {0, 1}$ .

Deducing Non-Zero Elements

We need the sum of $9$ values (each $0$ or $1$ ) to be $3$ .
Exactly $3$ elements must be $\pm 1$ (so $a_{ij}^{2} = 1$ ).
The remaining $6$ elements must be $0$ .

Choosing Positions

Number of ways to choose $3$ positions out of $9$ is $(3 9)$ .
$(3 9) = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$

Assigning Values

For each chosen position, $a_{ij}$ can be $1$ or $- 1$ ( $2$ choices).
Total ways to assign values $= 2 \times 2 \times 2 = 2^{3} = 8$ .

Final Calculation

Total matrices = (Ways to choose) $\times$ (Ways to assign)
Total $= 84 \times 8 = 672$

00:00 / 00:00

The Sigma Insight: Algebraic Operations on Matrices

Solution Diagram

Analyzing the Matrix Structure

We are tasked with finding the number of

3 \times 3

matrices

A

with entries

a_{ij} \in {- 1, 0, 1}

such that the trace of

A A^{T}

equals

3

The trace of

A A^{T}

is defined as the sum of its diagonal elements. Specifically, the

i

-th diagonal element

(A A^{T})_{ii}

is the dot product of the

i

-th row of

A

with itself.

Therefore, the trace is the sum of the squares of all nine elements in the matrix:

t r (A A^{T}) = i = 1 \sum 3 j = 1 \sum 3 a_{ij}^{2} = 3

Simplifying the Constraint

Given that each entry

a_{ij} \in {- 1, 0, 1}

, the square of any entry

a_{ij}^{2}

must be either

0

1

Since the sum of these nine squares must equal

3

, exactly three of the entries must have a square of

1

, while the remaining six entries must have a square of

0

This transforms the problem into a combinatorial selection task. We must choose

3

positions out of the

9

available slots in the matrix to be non-zero.

Combinatorial Calculation

The number of ways to choose

3

positions out of

9

is given by the binomial coefficient:

(3 9) = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84

For each of the

3

chosen positions, the entry

a_{ij}

can be either

1

- 1

(since both result in

a_{ij}^{2} = 1

Because there are

3

such positions, the number of ways to assign these values is:

2^{3} = 8

Final Calculation

To find the total number of such matrices, we multiply the number of ways to choose the positions by the number of ways to assign the values to those positions.

Total matrices =

84 \times 8 = 672

The total number of such matrices is 672.

Common Traps & FAQs

Why is

t r (A A^{T})

equal to the sum of squares of all elements? Can't we just multiply the matrices?

Great question! While you *could* perform the full matrix multiplication, it would be incredibly tedious. The property

t r (A A^{T}) = \sum_{i = 1}^{3} \sum_{j = 1}^{3} a_{ij}^{2}

is a powerful shortcut derived from the definition of matrix multiplication. In

A A^{T}

, the diagonal element

(A A^{T})_{ii}

is the dot product of the

i

-th row of

A

with the

i

-th column of

A^{T}

, which is the

i

-th row of

A

. This is exactly the sum of squares of the elements in that row. Summing over all

i

gives the sum of squares of all elements.

Why do we multiply by

2^{3}

at the end?

Remember, we determined that exactly three positions must be non-zero. For each of those three positions, the square of the entry must be

1

. Since the entries can be either

1

- 1

, and both

1^{2} = 1

and

(- 1)^{2} = 1

, each of those three positions has two possible values. Since there are three such positions, we have

2 \times 2 \times 2 = 2^{3} = 8

ways to assign the values.

Is it possible for the sum of squares to be something other than

3

The problem specifically constrains the sum of diagonal elements of

A A^{T}

to be

3

. If the problem had given a different value, say

5

, we would have needed

5

non-zero entries. The logic remains the same: identify the number of non-zero entries required, choose their positions, and then assign the values.

Animated Solution for Mathematics - Matrices and Determinants: The number of all $3 \times 3$ matrices A, with entries from the set ${- 1, 0, 1}$ such that the sum of the diagonal elements of $(A A^{T})$ is 3, is __________.

Enter Numerical Value:

Visualized Solution

Defining Matrix $A$

Trace of $A A^{T}$

Property of $t r (A A^{T})$

Setting up the Equation

Analyzing Possible Values

Deducing Non-Zero Elements

Choosing Positions

Assigning Values

Final Calculation

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Matrix Structure

Simplifying the Constraint

Combinatorial Calculation

Final Calculation

Similar Questions

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Enter Numerical Value:

Visualized Solution

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

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

Property of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } tr(AAT)

Setting up the Equation

Analyzing Possible Values

Deducing Non-Zero Elements

Choosing Positions

Assigning Values

Final Calculation

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Matrix Structure

Simplifying the Constraint

Combinatorial Calculation

Final Calculation

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Defining Matrix $A$

Trace of $A A^{T}$

Property of $t r (A A^{T})$

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Let $A = [a_{ij}]$ be a real matrix of order $3 \times 3$ , such that $a_{i 1} + a_{i 2} + a_{i 3} = 1$ , for $i = 1, 2, 3$ . Then, the sum of all the entries of the matrix $A^{3}$ is equal to: