Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Provide solution for rd sharma math class class 12 chapter Algebra of matrices exercise 4.3 question13

Provide solution for rd sharma math class class 12 chapter Algebra of matrices exercise 4.3 question 13

Answers (1)

Answer:

Hence, prove A B=B A=0_{3 \times 3}

Hint: matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: A=\left[\begin{array}{ccc}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right], B=\left[\begin{array}{ccc}a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2}\end{array}\right]

Consider,

A B=\left[\begin{array}{ccc}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right]\left[\begin{array}{lll}a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2}\end{array}\right] \\\\ \\ A B=\left[\begin{array}{ccc}0\left(a^{2}\right)+c(a b)+(-b)(a c) & (0)(a b)+(c)\left(b^{2}\right)+(-b)(b c) & (0)(a c)+c(b c)+(-b)\left(c^{2}\right) \\ (-c)\left(a^{2}\right)+(0)(a b)+a(a c) & (-c)(a b)+(0)\left(b^{2}\right)+a(b c) & (-c)(a c)+0(b c)+a\left(c^{2}\right) \\ b\left(a^{2}\right)+(-a)(a b)+0(a c) & (b)(a b)+(-a)\left(b^{2}\right)+0(b c) & (b)(a c)+(-a)(b c)+o\left(c^{2}\right)\end{array}\right]\) \\\\ \\A B=\left[\begin{array}{ccc}0+a b c-a b c & 0+b^{2} c-b^{2} c & 0+b c^{2}-b c^{2} \\ -a^{2} c+0+a^{2} c & -a b c+0+a b c & -a c^{2}+0+a c^{2} \\ a^{2} b-a^{2} b+0 & a b^{2}-a b^{2}+0 & a b c-a b c+0\end{array}\right]

A B=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] . . . . (i)

Again consider,

 

B A=\left[\begin{array}{ccc}a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2}\end{array}\right]\left[\begin{array}{ccc}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right] \\\\ \\ B A=\left[\begin{array}{ccc}\left(a^{2}\right)(0)+(a b)(-c)+(a c)(b) & a^{2}(c)+a b(0)+a c(-a) & a^{2}(-b)+a b(a)+a c(0) \\ a b(0)+b^{2}(-c)+b c(b) & a b(c)+b^{2}(0)+b c(-a) & a b(-b)+b^{2}(a)+b c(0) \\ a c(0)+b c(-c)+c^{2}(b) & a c(c)+b c(0)+c^{2}(-a) & a c(-b)+b c(a)+c^{2}(0)\end{array}\right] \\\\\\ B A=\left[\begin{array}{ccc}0-a b c+a b c & a^{2} c+0-a^{2} c & -a^{2} b+a^{2} b+0 \\ 0-b^{2} c+b^{2} c & a b c+0-a b c & -a b^{2}+a b^{2}+0 \\ 0-b c^{2}+b c^{2} & a c^{2}+0-a c^{2} & -a b c+a b c+0\end{array}\right] \\\\\\ B A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] \quad \ldots(i i)

From equation (i) & (ii)

A B=B A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] \\\\ A B=B A=0_{3 \times 3}

Hence, proved

Posted by

infoexpert22

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

PM Modi's Pariksha Pe Charcha 2025 to be held next month, registrations open till January 14
anam-khan

CBSE Board Exams 2025: Online portal for availing exemptions by CWSN students opens
anam-khan

Board Exams: States agree to equivalence; no question paper ‘jumbling’ from next year, says PARAKH CEO

Latest Question