Let p be an odd prime number and Tp be...

MathematicsTypes of MatricesJEE Advanced 2010Moderate

Comprehension Passage

Let

p

be an odd prime number and

T_{p}

be the following set of

2 \times 2

matrices :

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

Question 1:

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

(A)

(p - 1)^{2}

(B)

2 (p - 1)

(C)

(p - 1)^{2} + 1

(D)

2 p - 1

Answer:D

Question 2:

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)

(p - 1) (p - 2)

Answer:C

Question 3:

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

(A)

2 p^{2}

(B)

p^{3} - 5 p

(C)

p^{3} - 3 p

(D)

p^{3} - p^{2}

Answer:D

Visualized Solution (Hindi)

Understanding the Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T_{p}

Given set $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, \dots, p - 1}}$
Total number of matrices in $T_{p}$ is $p \times p \times p = p^{3}$
Determinant of $A$ is $det (A) = a^{2} - b c$

Symmetric Matrices in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T_{p}

For $A$ to be symmetric, $b = c$
$det (A) = a^{2} - b^{2} = (a - b) (a + b)$
$det (A) \equiv 0 (mod p) \Rightarrow a \equiv b (mod p)$ or $a \equiv - b (mod p)$

Counting Symmetric Solutions

Case 1: $a = b$ . Since $a \in {0, \dots, p - 1}$ , there are $p$ solutions.
Case 2: $a + b = p$ . Since $a, b \in {1, \dots, p - 1}$ , there are $p - 1$ solutions.
Total symmetric cases = $p + (p - 1) = 2 p - 1$

Skew-Symmetric Matrices

For $A$ to be skew-symmetric, $a = 0$ and $b = - c$
$det (A) = 0^{2} - b (- b) = b^{2}$
$det (A) \equiv 0 (mod p) \Rightarrow b^{2} \equiv 0 (mod p) \Rightarrow b = 0$
This leads to the zero matrix ( $a = b = c = 0$ ), already counted in the symmetric case.
Final answer for Q1: $2 p - 1$

Trace Condition in Question 2

$trace (A) = 2 a$
$trace (A) \neq \equiv 0 (mod p) \Rightarrow 2 a \neq \equiv 0 (mod p) \Rightarrow a \neq = 0$
Choices for $a \in {1, 2, \dots, p - 1}$ is $p - 1$

Determinant Condition .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b c \equiv a^{2}

$det (A) = a^{2} - b c \equiv 0 (mod p) \Rightarrow b c \equiv a^{2} (mod p)$
Since $a \neq = 0$ , $a^{2} \neq \equiv 0 (mod p)$ , so $b \neq = 0$ and $c \neq = 0$
For a fixed $a$ , and each $b \in {1, \dots, p - 1}$ , there is a unique $c = a^{2} b^{- 1} (mod p)$

Total Count for Question 2

Number of choices for $a = p - 1$
Number of pairs $(b, c)$ for each $a = p - 1$
Total matrices = $(p - 1) \times (p - 1) = (p - 1)^{2}$
Final answer for Q2: $(p - 1)^{2}$

Total Matrices with .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } det (A) Divisible by .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p

Total matrices in $T_{p} = p^{3}$
To find $det (A) \neq \equiv 0$ , we calculate $det (A) \equiv 0$ and subtract.
Case 1: $a = 0$ . $det (A) = - b c \equiv 0 (mod p) \Rightarrow b = 0$ or $c = 0$
Number of ways for $a = 0$ is $p + p - 1 = 2 p - 1$

Final Calculation for Question 3

Case 2: $a \neq = 0$ . From Q2, number of ways is $(p - 1)^{2}$
Total matrices with $det (A) \equiv 0 (mod p) = (2 p - 1) + (p - 1)^{2}$
$= 2 p - 1 + p^{2} - 2 p + 1 = p^{2}$
Number of matrices with $det (A) \neq \equiv 0 (mod p) = p^{3} - p^{2}$
Final answer for Q3: $p^{3} - p^{2}$

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Conceptually Similar Problems

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Let

A

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Question 1:

The number of matrices in $A$ is

(A)

(B)

(C)

(D)

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

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(B)

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(C)

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(D)

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Let A and B be

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sin θ = cos ϕ

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(1)

(2)

(3)

(4)

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Let $ω \neq = 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $1 ω ω^{2} a 1 ω b c 1$ where each of $a, b$ and $c$ is either $ω$ or $ω^{2}$ . Then the number of distinct matrices in the set $S$ is

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Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$ . If $P^{T}$ denotes the transpose of $P$ , then $M^{2} N^{2} (M^{T} N)^{- 1} (M N^{- 1})^{T}$ is equal to

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Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $t r (A)$ , the sum of diagonal entries of $A$ . Assume that $A^{2} = I$ . Statement-1: If $A \neq = I$ and $A \neq = - I$ , then $d e t (A) = - 1$ Statement-2: If $A \neq = I$ and $A \neq = - I$ , then $t r (A) \neq = 0$ .

(A)

Statement-1 is false, Statement-2 is true

(B)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(C)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

(D)

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MathematicsTypes of MatricesJEE Main 2010Moderate

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

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(C)

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Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

(A)

(a)

2^{10}

(B)

(b)

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Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$ , non zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

* Multiple Correct Options

(A)

Y^{3} Z^{4} - Z^{4} Y^{3}

(B)

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(C)

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(D)

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Let $ω$ be a complex cube root of unity with $ω \neq = 1$ and $P = [p_{ij}]$ be an $n \times n$ matrix with $p_{ij} = ω^{i + j}$ . Then $P^{2} \neq = 0$ , when $n =$

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57

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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 A in Tp​ such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that the trace of A is not divisible by p but det(A) is divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that det(A) is not divisible by p is

Visualized Solution (Hindi)

Understanding the Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tp​

Symmetric Matrices in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tp​

Counting Symmetric Solutions

Skew-Symmetric Matrices

Trace Condition in Question 2

Determinant Condition .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } bc≡a2

Total Count for Question 2

Total Matrices with .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } det(A) Divisible by .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p

Final Calculation for Question 3

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