Question
Prove that the diagonal elements of a skew symmetric matrix are all zero.
✓
Answer
Let $A$ be a skew-symmetric matrix. Then by definition $A^{\prime}=-A$
or the $(i, j)^{\text {th }}$ element of $A^{\prime}=$ the $(i, j)^{\text {th }}$ element of $(-A)$
or the $(j, i)^{\text {th }}$ element of $A=-$ the $(i, j)^{\text {th }}$ element of $A$
For the diagonal elements $i=j$ or the $(i, j)$ the element of $A=-$ the $(i, j)^{\text {th }}$ element of $A$
or the $(i, i)^{\text {th }}$ element of $A=0$
Hence the diagonal elements are all zero.
or the $(i, j)^{\text {th }}$ element of $A^{\prime}=$ the $(i, j)^{\text {th }}$ element of $(-A)$
or the $(j, i)^{\text {th }}$ element of $A=-$ the $(i, j)^{\text {th }}$ element of $A$
For the diagonal elements $i=j$ or the $(i, j)$ the element of $A=-$ the $(i, j)^{\text {th }}$ element of $A$
or the $(i, i)^{\text {th }}$ element of $A=0$
Hence the diagonal elements are all zero.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Evaluate: $(9+23) \bmod 12$
→Three car dealers, say A, B and C deals in three cars, Sedan, cars Hatchback cars and SUV cars. The sales figure of 220 and 2021 showed that dealer A sold 120 Hatch back, 50 Sedan and 10 SUV cars in 2020 and 300 Hatchback, 150 Sedan and 20 SUV in 2021. Dealer B sold 100 Hatchback, 30 Sedan, 5 SUV cars in 2020 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2021. Dealer C sold 90 Hatchback, 40 Sedan and 2 SUV cars in 2020 and 100 Hatchback, 60 Sedan and 5 SUV cars in 2021
Q.1. Find the total number of cars sold in two given years, by each dealer.
Q.2. If each dealer receive profit of ₹ $ 50,000$ on sale of a Hatch car, ₹ $ 1,00,000$ on sale of a sedan car and ₹ $ 1,50,000$ on a sale of a SUV car, then find the amount of profit received in year 2021 by each dealer.
→Q.1. Find the total number of cars sold in two given years, by each dealer.
Q.2. If each dealer receive profit of ₹ $ 50,000$ on sale of a Hatch car, ₹ $ 1,00,000$ on sale of a sedan car and ₹ $ 1,50,000$ on a sale of a SUV car, then find the amount of profit received in year 2021 by each dealer.
Evaluate: $\int\left(\frac{1-x}{1+x^2}\right)^2 e^x d x$
→Find $\varphi(n)$ for $n =3$
→A telephone switch board handles 600 calls on the average during a rush hour. The board can make a maximum of 20 connections per minute. Use poisson distribution to estimate the probability the board will be over taxed during any given minute. $\left[e^{-1}=0.00004539\right]$
→Find the differential equation representing the family of curves $V=\frac{A}{r}+B$, where A and B are arbitrary constants.
→Amrita bought a car worth ₹ $12,50,000$ and makes a down payment of ₹ $3,00,000.$ The balance amount is to be paid in 4 years by equal monthly instalments at an interest rate of $15\%$ p.a. Find the EMI that Amrita has to pay for the car. $\left\{\right.$Given $\left.\left.(1.0125)^{-48}=0.5508565\right)\right\}$
→Find $a -_{ m } b$, if $a =11, b=43$ and $m =7$
→Find the general solution of the differential equation $\frac{d y}{d x}=e^{x+y}$
→P,Q and R jointly invested a total of 64000 in a business. P invested 6000 more than Q and Q
invested 5000 more than R. Find the ratio of their capital?
invested 5000 more than R. Find the ratio of their capital?