The given problem is to find the value of A³, where A is a 2×2 matrix given by A = (a b c d).

Given condition:
The given condition is a*d - b*c = 1.

Solution:

Matrix Multiplication:
To find the value of A³, we need to multiply matrix A three times.

First Multiplication:
A * A = (a b c d) * (a b c d)
= (a*a + b*c a*b + b*d c*a + d*c c*b + d*d)
= (a² + bc ab + bd ac + cd bc + d²)
= (a² + bc ab + bd ac + cd bc + d²)

Second Multiplication:
(A * A) * A = (a² + bc ab + bd ac + cd bc + d²) * (a b c d)
= (a² + bc)*a + (ab + bd)*c (a² + bc)*b + (ab + bd)*d (ac + cd)*a + (bc + d²)*c (ac + cd)*b + (bc + d²)*d
= a³ + abc + acd + bcd + bd² a²b + bcd + abd + b²d a²c + bcd + ac² + c²d abc + bcd + acd + bd³

Third Multiplication:
(A * A * A) = (a³ + abc + acd + bcd + bd² a²b + bcd + abd + b²d a²c + bcd + ac² + c²d abc + bcd + acd + bd³) * (a b c d)
= (a³ + abc + acd + bcd + bd²)*a + (a²b + bcd + abd + b²d)*c (a²c + bcd + ac² + c²d)*a + (abc + bcd + acd + bd³)*c
= a⁴ + a²bc + a²cd + abcd + abd² + a²bc + bcd² + ac²d + bd²c + ac³ + bcd² + a²c² + c³d + abc² + bcd² + acd³ + bd⁴
= a⁴ + 2a²bc + a²cd + abcd + abd² + bcd² + ac²d + bd²c + ac³ + a²c² + c³d + abc² + acd³ + bd⁴

Simplifying the expression:
Now, let's simplify the expression by combining the like terms.

A³ = a⁴ + 2a²bc + a²cd + abcd + abd² + bcd² + ac²d + bd²c + ac³ + a²c² + c³d + abc² + acd³ + bd⁴

Hence, A³ = a⁴ + 2a