Final answer:
To determine if the matrix N¹⁰⁰ is invertible, we calculate the determinant of N and then raise it to the power of 100. MATLAB's det function simplifies this process, and if det(N) is not zero, N¹⁰⁰ is invertible without needing to reconsider the calculation.
Step-by-step explanation:
The student is interested in learning about the properties of matrices and their determinants, in the context of MATLAB programming. We begin by entering the given matrix N in MATLAB:
N = [-0.003 0.02 0; 0.1 1 0; 0 0 0.015];
Next, to find the determinant of the matrix N raised to the power of 100, N¹⁰⁰, we can use MATLAB's det function as:
detN100 = det(N)^100;
If the determinant of N is not zero, then the matrix is invertible, and since raising a non-zero number to any power (except zero) results in a non-zero number, N¹⁰⁰ would also be invertible. To compute the determinant of the original matrix N:
detN = det(N);
Now, if you calculate the determinant of N¹⁰⁰ by hand, using the property of determinants that the determinant of a matrix raised to a power is equal to the determinant of the matrix raised to that power, you would get the same result as above (det(N)¹⁰⁰), assuming the determinant is non-zero.
The conclusion is that if det(N) is non-zero, we can be confident that N¹⁰⁰ is indeed invertible. There is no need to reconsider the answer, as the determinant calculated by hand and using MATLAB should yield the same result in this case.