If A = ⎡⎢⎣2 -3 5⎤⎥⎦3 2 -1⎤⎥⎦, find A⁻¹. Use it to solve the system of equations: 2x - y + 5z = 11; 3x + 2y - z = 9; x + y - 2z = 7.

Co(A) = ⎡⎢⎣0 2 1⎤⎥⎦-1 -5 -1⎤⎥⎦-1 -7 0⎤⎥⎦
We know that Adj(A) = Transpose of Co(A)
∴Adj(A) = ⎡⎢⎣0 -1 -1⎤⎥⎦2 -5 -7⎤⎥⎦1 -1 0⎤⎥⎦
Since, A⁻¹ = (1/|A|)Adj(A) and |A| = 9
∴A⁻¹ = ⎡⎢⎣0 1/9 -1/9⎤⎥⎦2/9 -5/9 -7/9⎤⎥⎦1/9 -1/9 0⎤⎥⎦
Now, Ax = b ⇒ x = A⁻¹b
⇒⎡⎢⎣xyz⎤⎥⎦ = ⎡⎢⎣0 1/9 -1/9⎤⎥⎦2/9 -5/9 -7/9⎤⎥⎦1/9 -1/9 0⎤⎥⎦ ⎡⎢⎣1197⎤⎥⎦
Therefore,⎡⎢⎣xyz⎤⎥⎦ = ⎡⎢⎣123⎤⎥⎦
Hence, x = 1, y = 2 and z = 3.