If A = ⎡⎢⎣2 -3 5⎤⎥⎦, then find A⁻¹ and hence solve the system of linear equations 2x - y + 5z = 11, 3x + 2y - z = 9 and x + y - 2z = -9

|A| = det(A) = ∣∣∣∣2 -3 5∣∣∣∣ = 2(8+4) + 3(12+4) + 5(-9-2) = 24 + 48 - 55 = 17

Now, finding the cofactors of the matrix A:
A₁₁ = ∣∣∣2 1∣∣∣ = 17
A₁₂ = -∣∣∣3 1∣∣∣ = -23
A₁₃ = ∣∣∣3 2 1∣∣∣ = -1
A₂₁ = -∣∣∣-3 5∣∣∣ = -13
A₂₂ = ∣∣∣2 5 1∣∣∣ = -13
A₂₃ = -∣∣∣2 -3∣∣∣ = 5
A₃₁ = ∣∣∣-3 5∣∣∣ = -7
A₃₂ = -∣∣∣2 5 3∣∣∣ = -17
A₃₃ = ∣∣∣2 -3∣∣∣ = 17

Adjoint of matrix A will be given by:
adj(A) = ⎡⎢⎣17 -23 -1
-13 -13 5
-7 -17 17⎤⎥⎦ᵀ = ⎡⎢⎣17 -13 -7
-23 -13 -17
-1 5 17⎤⎥⎦

∴ A⁻¹ = (1/|A|) adj(A) = (1/17) ⎡⎢⎣17 -13 -7
-23 -13 -17
-1 5 17⎤⎥⎦

The system of equations can be reduced in the matrix form AX = B as
⎡⎢⎣2 -3 5⎤⎥⎦⎡⎢⎣x
y
z⎤⎥⎦ = ⎡⎢⎣11
9
-9⎤⎥⎦

∴ ⎡⎢⎣x
y
z⎤⎥⎦ = A⁻¹⎡⎢⎣11
9
-9⎤⎥⎦ = (1/17)⎡⎢⎣17 -13 -7
-23 -13 -17
-1 5 17⎤⎥⎦⎡⎢⎣11
9
-9⎤⎥⎦ = (1/17)⎡⎢⎣187 -117 -77
-253 -117 -153
-11 45 -153⎤⎥⎦ = ⎡⎢⎣11
7
-3⎤⎥⎦

Therefore x = 1, y = 2, z = 3 is the solution to the system of equations.