Mathematics 2000 JAMB Past Questions - Question 15: a matrix P(a b)(c d) is such that Pͭ =-p where Pͭ is the transpose of p. if b=1, then p is

Correct Answer

A

Explanation

If you have a matrix P given by:P = | a b | | c d |To find the matrix P given that Pͭ = -P and b = 1, we can start by writing the transpose of P:Pͭ = (a c)      (b d)Since Pͭ = -P, we have:(a c) = -(a b)(b d)   -(c d)From this, we can see that a = -a, b = -c, c = -b, and d = -d. Since b = 1, we can substitute it into the equations:-a = -a1 = -c-c = -1d = -dFrom the first equation, we can see that a = 0. From the second equation, we have c = -1. From the third equation, we have c = 1. Finally, from the fourth equation, we have d = 0.Therefore, the matrix P is:P = (0 1)      (-1 0)And it's known that the transpose of P (denoted as P^T) is equal to the negative of P, then you can write this relationship as:P^T = -PIn this case, you are given that b = 1. So, P looks like this:P = | a 1 | | c d |Now, let's find the transpose of P, P^T:P^T = | a c | | 1 d |According to the given condition, P^T should be equal to -P:| a c | = - | a 1 | | c d |Now, let's equate the corresponding elements:a = -a (1) c = 1 (2) c = -c (3) d = -1 (4)From equation (1), we see that a must be 0 because -a = a implies a = 0.From equation (2), we know that c = 1.Now, we can use equation (3) to see that c = -c. This means that c must be 0.Finally, from equation (4), we find that d = -1.So, the matrix P that satisfies the condition P^T = -P when b = 1 is:P = | 0 1 | | 0 -1 |