Question

Determine the condition number of the following matrices. Comment on whether or not small errors in data of each matrix A can result in large errors in the solution of Ax=b. a [ 1 3 ]

[ 2 4 ]
b [ 2 -3 ]
[ -2 1 ]
c [ 1 -3 ]
[ -2 6 ]
d [ 1/3 1/4 1/5 ]
[ 1/4 1/5 1/6 ]
[ 1/5 1/6 1/7 ]

Answer 1

The condition numbers for matrices A, B, and C are approximately 7.99, 16.53, and 63.97, respectively. Matrix D has a condition number of about 72.45. Small errors in data of matrices with higher condition numbers, like C and D, can result in large errors in the solution of Ax=b.

Step-by-step explanation:

The condition number of a matrix measures how sensitive the solution of a linear system is to small changes in the input data. It is calculated as the product of the matrix norm and the norm of its inverse.

The higher the condition number, the more ill-conditioned the matrix, indicating increased sensitivity to input changes.

Matrix A has a condition number of approximately 7.99, which is relatively low. This suggests that small errors in the data of matrix A are less likely to result in large errors in the solution of Ax=b.

Matrix B has a higher condition number of about 16.53, indicating a moderate level of ill-conditioning. Errors in the input data of matrix B can lead to somewhat larger errors in the solution compared to matrix A.

Matrix C has a condition number of approximately 63.97, signifying significant ill-conditioning. Small errors in the data of matrix C can result in relatively large errors in the solution of Ax=b.

Matrix D has the highest condition number among the given matrices, about 72.45. This makes matrix D highly ill-conditioned, and small errors in its data are likely to cause substantial errors in the solution of the linear system Ax=b.

In practical terms, choosing matrices with lower condition numbers is desirable to ensure numerical stability in solving linear systems.

Answer 2

The condition number for the given matrices is calculated to determine whether small errors can result in large errors in the solution.

Step-by-step explanation:

The condition number of a matrix can be used to determine how sensitive the solution of the matrix equation Ax=b is to small errors in the data. The condition number is given by the product of the norm of the matrix A and the norm of its inverse.

Let's calculate the condition number for the given matrices:

1. Matrix A: The condition number is 4/0 = ∞, indicating that small errors in the data can result in large errors in the solution.
2. Matrix B: The condition number is 7/0.0211 ≈ 332, indicating that small errors in the data can still result in relatively large errors in the solution.
3. Matrix C: The condition number is 9/0.0211 ≈ 427, indicating that small errors in the data can also result in relatively large errors in the solution.
4. Matrix D: The condition number is 49/0.00114 ≈ 42,982, indicating that small errors in the data can result in extremely large errors in the solution.

In summary, matrices A, B, C, and D have high condition numbers, indicating that small errors in the data of these matrices can result in large errors in the solution of Ax=b.