import numpy
from numpy.linalg import LinAlgError

n = 3
A = numpy.random.randint(low=1, high=10, size=(n, n))
print("Matrix A: \n", A)
try:
    Q, R = numpy.linalg.qr(A)
    print("The Orthogonal Matrix Q: \n", Q)
    print("The Upper Triangular Matrix R: \n", R)
except LinAlgError:
    print("The QR Decomposition of The Matrix A Failed")

R_inverse = numpy.linalg.inv(R)
Q_transpose = numpy.transpose(Q)
A_inverse = numpy.matmul(R_inverse, Q_transpose)

try:
    if numpy.allclose(A_inverse, numpy.linalg.inv(A)):
        print("The inverse of A: \n", A_inverse)
except LinAlgError:
    print("The matrix A is not invertible")

Here, A is a 3×3 square matrix that contains random integers. We are using the numpy.randopm.randint() function for generating random integers. The generated random integers will be within the range [low, high). And the size=(n, n) argument specifies that the created matrix will have n rows and n columns.

We are using the function numpy.linalg.qr() to perform QR decomposition of the matrix A. If the QR decomposition fails, the function will raise a LinAlgError. Otherwise, we will get one orthogonal matrix Q and one upper triangular matrix R.

Now, we can calculate R^-1Q^T to get the inverse of the matrix A.

R_inverse = numpy.linalg.inv(R)
Q_transpose = numpy.transpose(Q)
A_inverse = numpy.matmul(R_inverse, Q_transpose)

try:
    if numpy.allclose(A_inverse, numpy.linalg.inv(A)):
        print("The inverse of A: \n", A_inverse)
except LinAlgError:
    print("The matrix A is not invertible")

Here, we are using the numpy.allclose() function to check whether R-1QT is indeed equal to the inverse of the matrix A.

The output of the given program will be:

Matrix A: 
 [[6 1 9]
 [9 6 4]
 [9 3 7]]
The Orthogonal Matrix Q: 
 [[-0.42640143  0.58693919 -0.6882472 ]
 [-0.63960215 -0.73367398 -0.22941573]
 [-0.63960215  0.34238119  0.6882472 ]]
The Upper Triangular Matrix R: 
 [[-14.07124728  -6.18282077 -10.87323653]
 [  0.          -2.78796113   4.74442508]
 [  0.           0.          -2.29415734]]
The inverse of A: 
 [[-0.33333333 -0.22222222  0.55555556]
 [ 0.3         0.43333333 -0.63333333]
 [ 0.3         0.1        -0.3       ]]