If matrix A = [ 0 1 , -1 1]

find the value of p and q such that (pl+qA)^2 =A.

looks good to me.

To find the value of p and q such that (pl+qA)^2 = A, we can follow these steps:

Step 1: Expand the square on the left side of the equation:
(pl + qA)(pl + qA) = A

Step 2: Apply matrix multiplication rules:
p^2l^2 + 2plqA + q^2A^2 = A

Step 3: Simplify the equation using the given matrix A:
p^2l^2 + 2plqA + q^2A^2 = [0 1, -1 1]

Step 4: Break down the equation into individual components:
p^2l^2 = 0
2plqA = [0 1, -1 1]
q^2A^2 = [0 1, -1 1]

Step 5: Solve the first component:
Since p^2l^2 = 0, this implies that either p or l must be zero. Let's consider two cases:

Case 1: p = 0
If p = 0, then the equation becomes: 0 = 0. In this case, any value of q would satisfy the equation.

Case 2: l = 0
If l = 0, then the equation becomes: p^2(0)^2 = 0. This implies that p can be any real number.

Step 6: Solve the second and third components:
To solve for q and find compatible values of p and q, we need to consider the second component: 2plqA = [0 1, -1 1], and the third component: q^2A^2 = [0 1, -1 1].

By comparing the second component ([0 1, -1 1]) to qA, we can see that q = 1/2. Then, substitute this value back into the third component: (1/2)^2A^2 = [0 1, -1 1]. Since A^2 = [0 2, -2 0], we get:

(1/4)[0 2, -2 0] = [0 1, -1 1]

This is true for q = 1/2, regardless of the values of p and l.

In conclusion, for any value of p (including p = 0) and q = 1/2, the equation (pl+qA)^2 = A holds true.

note that pI+qA =

(p q)
(-q p+q)
and that means that if you square that you get
(p^2-q^2 q(2p+q))
(-q(2p+q) p(p+2q))
so if you set that equal to A, you have
p^2-q^2 = 0
q(2p+q) = 1
p(p+2q) = 1
now just solve that for p and q.

So i got

p = +- q
Then
q = +-1/root3 , then p also same , when p = q

When p = - q, then q= +- i
Hence p = - + i

AM I RIGHT OR WRONG @oobleck