Suppose A and B are non-zero matrices such that A^2=0 and B^3=0(But B^2 does not =0) find in simplst form:
a)(A+B)^2
b)(A+B)^3
c)(A+B)^4
d)(A+B)^5
e)(A+B)^6
To find the expressions (A+B)^2, (A+B)^3, (A+B)^4, (A+B)^5, and (A+B)^6, we can use the binomial theorem along with the given properties of matrices A and B.
The binomial theorem states that for any two matrices A and B and any positive integer n, we can expand (A + B)^n as the sum of terms of the form C(n, k) * A^(n-k) * B^k, where C(n, k) represents the binomial coefficient (also known as the combination) and is calculated as C(n, k) = n! / (k! * (n-k)!).
Let's evaluate each expression step-by-step:
a) (A+B)^2:
Using the binomial theorem, we have (A+B)^2 = C(2, 0) * A^2 * B^0 + C(2, 1) * A^1 * B^1 + C(2, 2) * A^0 * B^2.
Since A^2 = 0 and B^0 = I (the identity matrix), the expression simplifies to 0 + 2 * A * B + 0 = 2AB.
b) (A+B)^3:
Similarly, we have (A+B)^3 = C(3, 0) * A^3 * B^0 + C(3, 1) * A^2 * B^1 + C(3, 2) * A^1 * B^2 + C(3, 3) * A^0 * B^3.
Since A^2 = 0 and B^3 = 0, the expression simplifies to 0 + 0 + 3 * A * B^2 + 0 = 3AB^2.
c) (A+B)^4:
Following the same pattern, (A+B)^4 = C(4, 0) * A^4 * B^0 + C(4, 1) * A^3 * B^1 + C(4, 2) * A^2 * B^2 + C(4, 3) * A^1 * B^3 + C(4, 4) * A^0 * B^4.
Since A^2 = 0 and B^3 = 0, the expression simplifies to 0 + 0 + 0 + 4 * A * B^3 + 0 = 4AB^3.
d) (A+B)^5:
Again, using the binomial theorem, (A+B)^5 = C(5, 0) * A^5 * B^0 + C(5, 1) * A^4 * B^1 + C(5, 2) * A^3 * B^2 + C(5, 3) * A^2 * B^3 + C(5, 4) * A^1 * B^4 + C(5, 5) * A^0 * B^5.
Since A^2 = 0 and B^3 = 0, the expression simplifies to 0 + 0 + 0 + 0 + 5 * A * B^4 + 0 = 5AB^4.
e) (A+B)^6:
Following the pattern, (A+B)^6 = C(6, 0) * A^6 * B^0 + C(6, 1) * A^5 * B^1 + C(6, 2) * A^4 * B^2 + C(6, 3) * A^3 * B^3 + C(6, 4) * A^2 * B^4 + C(6, 5) * A^1 * B^5 + C(6, 6) * A^0 * B^6.
Since A^2 = 0 and B^3 = 0, the expression simplifies to 0 + 0 + 0 + 0 + 0 + 6 * A * B^5 + 0 = 6AB^5.
So, the simplest form of the given expressions are:
a) (A+B)^2 = 2AB
b) (A+B)^3 = 3AB^2
c) (A+B)^4 = 4AB^3
d) (A+B)^5 = 5AB^4
e) (A+B)^6 = 6AB^5