Let A be the set of all 3 x 3 symmetric...

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2009Moderate

Comprehension Passage

Let

A

be the set of all

3 \times 3

symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Question 1:

The number of matrices in $A$ is

(A)

(B)

(C)

(D)

Answer:A

Question 2:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

(A)

less than 4

(B)

at least 4 but less than 7

(C)

at least 7 but less than 10

(D)

at least 10

Answer:B

Question 3:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

(A)

(B)

more than 2

(C)

(D)

Answer:B

Visualized Solution (Hindi)

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

A symmetric $3 \times 3$ matrix has the form $A = a d e d b f e f c$
Total entries = $9$ . Given: five $1$ s and four $0$ s.
Symmetry implies off-diagonal entries $(d, e, f)$ appear in pairs.
Let $k$ be the number of $1$ s on the diagonal. Then $5 - k$ must be even.
Possible values for $k$ : $1$ or $3$ .

Case 1: One .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1 on the Diagonal

Case 1: $k = 1$ ( $1$ on diagonal, $2$ zeros on diagonal)
Ways to choose diagonal: $(1 3) = 3$
Remaining $1$ s = $4$ , which forms $2$ off-diagonal pairs.
Ways to choose off-diagonal pairs: $(2 3) = 3$
Total matrices for Case 1: $3 \times 3 = 9$

Case 2: Three .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1 s on the Diagonal

Case 2: $k = 3$ (all $3$ on diagonal are $1$ s)
Ways to choose diagonal: $(3 3) = 1$
Remaining $1$ s = $2$ , which forms $1$ off-diagonal pair.
Ways to choose off-diagonal pairs: $(1 3) = 3$
Total matrices for Case 2: $1 \times 3 = 3$
Total matrices in A = $9 + 3 = 12$

Condition for Unique Solution

System $A x = b$ has a unique solution if $∣ A ∣ \neq = 0$ .
Determinant $∣ A ∣ = ab c + 2 d e f - a f^{2} - b e^{2} - c d^{2}$
Since entries $\in {0, 1}$ , $∣ A ∣ = ab c + 2 d e f - a f - b e - c d$

Counting Unique Solutions

Checking $12$ matrices:
From Case 1 ( $k = 1$ ): $6$ matrices have $∣ A ∣ = - 1$ (e.g., $a = 1, e = 1, f = 1, b = c = d = 0$ )
From Case 2 ( $k = 3$ ): All $3$ matrices have $∣ A ∣ = 0$
Total matrices with unique solution = $6$
This fits the range: at least 4 but less than 7

Condition for Inconsistency

System is inconsistent if $∣ A ∣ = 0$ and at least one of $D_{x}, D_{y}, D_{z} \neq = 0$ .
We check the $6$ matrices where $∣ A ∣ = 0$ .
Example: $A = 110110001$ gives $x + y = 1$ and $x + y = 0$ , which is inconsistent.

Final Count for Inconsistency

Out of $6$ matrices with $∣ A ∣ = 0$ :
Inconsistent cases: $4$
Consistent (Infinite solutions) cases: $2$
Total inconsistent matrices = $4$
Answer: more than 2

00:00 / 00:00

Conceptually Similar Problems

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T_{p}

be the following set of

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matrices :

T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}

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(D)

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The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

(A)

(p - 1) (p^{2} - p + 1)

(B)

p^{3} - (p - 1)^{2}

(C)

(p - 1)^{2}

(D)

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The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

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(D)

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, and

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(B)

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Question 2:

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(B)

(C)

(D)

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Consider the set $A$ of all determinants of order 3 with entries 0 or 1 only. Let $B$ be the subset of $A$ consisting of all determinants with value 1. Let $C$ be the subset of $A$ consisting of all determinants with value $- 1$ . Then

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(a)

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(B)

(b)

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(C)

(c)

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(D)

(d)

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Let $A = [1324]$ and $B = [a 0 0 b]$ , $a, b \in N$ . Then

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(B)

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Let $a, b, c$ be the real numbers. Then following system of equations in $x, y$ and $z$ $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1$ , $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1$ , $- \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1$ has

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(b) unique solution

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Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices in A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ has a unique solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ is inconsistent, is

Visualized Solution (Hindi)

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

Case 1: One .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1 on the Diagonal

Case 2: Three .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1s on the Diagonal

Condition for Unique Solution

Counting Unique Solutions

Condition for Inconsistency

Final Count for Inconsistency

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The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

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Case 2: Three $1$ s on the Diagonal

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct solutions, is

The number of $3 \times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$ , is

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

Let $A = [1324]$ and $B = [a 0 0 b]$ , $a, b \in N$ . Then

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct solutions, is

The number of $3 \times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$ , is

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

Let $A = [1324]$ and $B = [a 0 0 b]$ , $a, b \in N$ . Then