Show that matrix [[0, 0, 1], [0, 1, 0], [0, 0, 0]] is invertible. Find its inverse by adjoint method.
Understand the Problem
The question is asking to show that the matrix [[0, 0, 1], [0, 1, 0], [0, 0, 0]] is invertible and to find its inverse using the adjoint method. This involves calculating the determinant and using it to find the inverse.
Answer
The matrix is not invertible; its determinant is $|A| = 0$.
Answer for screen readers
The matrix
$$ A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix} $$
is not invertible because its determinant is zero.
Steps to Solve
- Calculate the Determinant
To check if the matrix is invertible, we must first find its determinant. The matrix is
$$ A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix} $$
We can compute the determinant using the formula for 3x3 matrices:
$$ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) $$
Here, ( a = 0, b = 0, c = 1, d = 0, e = 1, f = 0, g = 0, h = 0, i = 0 ).
Substituting the values, we find:
$$ |A| = 0(10 - 00) - 0(00 - 00) + 1(00 - 10) = 0 $$
- Determine Invertibility
Since the determinant ( |A| = 0 ), the matrix is not invertible. We must show this and conclude that the matrix has no inverse.
- Conclusion
Given that the determinant is zero, we cannot use the adjoint method to find an inverse for this matrix because an inverse does not exist for matrices with a determinant of zero.
The matrix
$$ A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix} $$
is not invertible because its determinant is zero.
More Information
A matrix is invertible if and only if its determinant is non-zero. In this case, we showed that the determinant is zero, confirming the matrix cannot have an inverse.
Tips
- Miscalculating the determinant of a matrix can lead to incorrect conclusions about invertibility. It's important to double-check the arithmetic used in calculating determinants.
- Assuming any 3x3 matrix is invertible without determining its determinant.