SATHEE: Matrices and Determinants 1 Question 19

Matrices and Determinants 1 Question 19

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

(a) $(p - 1)^{2}$

(b) $2 (p - 1)$

(d) $2 p - 1$

NOTE: The trace of a matrix is the sum of its diagonal entries.

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Answer:

Correct Answer: 19. (d)

Solution:

Given, $A = [\begin{matrix} a & b \\ c & a \end{matrix}]$ , $a, b, c \in 0, 1, 2, \dots, p - 1$

If $A$ is skew-symmetric matrix, then $a = 0, b = - c$

$∴ | A | = - b^{2}$

Thus, $P$ divides $| A |$ , only when $b = 0$ …(i)

Again, if $A$ is symmetric matrix, then $b = c$ and

$| A | = a^{2} - b^{2}$

Thus, $p$ divides $| A |$ , if either $p$ divides $(a - b)$ or $p$ divides $(a + b)$ .

$p$ divides $(a - b)$ , only when $a = b$ ,

i.e. $a = b \in 0, 1, 2, \dots, (p - 1)$

i.e. $p$ choices …(ii)

$p$ divides $(a + b)$

$\Rightarrow p$ choices, including $a = b = 0$ included in Eq. (i).

$∴$ Total number of choices are $(p + p - 1) = 2 p - 1$

Matrices and Determinants 1 Question 19

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

Answer:

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Matrices and Determinants 1 Question 19

19. The number of A in Tp such that A is either symmetric or skew-symmetric or both and det(A) is divisible by p is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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