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Inverse of diagonal matrix
A diagonal matrix with the given values: 2,3,4 as follows.
$$ 𝐀 = \text{diag\\{2, 3, 4\\}} = \begin{bmatrix} 2 & 0 & 0 \\\ 0 & 3 & 0 \\\ 0 & 0 & 4 \end{bmatrix} $$The augmented matrix is constructed and perform elementary row operations as:
$$ 𝐀|𝐈 = \begin{bmatrix} 2 & 0 & 0 & | 1 & 0 & 0 \\\ 0 & 3 & 0 & | 0 & 1 & 0 \\\ 0 & 0 & 4 & | 0 & 0 & 1 \end{bmatrix} \overset{1/2r1,1/3r2, 1/4r3}{⇢} \begin{bmatrix} 1 & 0 & 0 & | 1/2 & 0 & 0 \\\ 0 & 1 & 0 & | 0 & 1/3 & 0 \\\ 0 & 0 & 1 & | 0 & 0 & 1/4 \end{bmatrix} $$From the results, the inverse matrix 𝐀⁻¹ of a diagonal matrix is also a diagonal matrix. And its elements on the main diagonal are the reciprocal of elements on the original diagonal, where the element cannot be 0.
$$ 𝐀⁻¹ = \begin{bmatrix} 1/2 & 0 & 0 \\\ 0 & 1/3 & 0 \\\ 0 & 0 & 1/4 \end{bmatrix} = \text{diag\\{1/2, 1/3, 1/4\\}} $$Inverse of a scalar matrix
If all elements on the diagonal are the same, the matrix is a schalar matrix. 𝚲 = diag{λ, …, λ}, λ≠0.
Its inverse matrix is a scalar matrix where the elements on the diagonal are all 1/λ, 𝚲⁻¹ = diag{λ⁻¹, …, λ⁻¹}, λ≠0.
If λ = 1, the scalar matrix is identity matrix 𝐈. So the inverse of a identity matrix is itself: 𝐈⁻¹ = 𝐈.
Inverse of the elementary matrix
The elementary matrix is definitely invertible. So their inverse matrices must exist.
Inverse of a row-switching matrix
For a row-switching matrix (permutation matrix) switching the first 2 rows of the identity matrix:
$$ 𝐅₁ = \begin{bmatrix} 0 & 1 & 0 \\\ 1 & 0 & 0 \\\ 0 & 0 & 1 \end{bmatrix} $$Write the augmented matrix:
$$ 𝐅₁|𝐈 = \begin{bmatrix} 0 & 1 & 0 &| 1 & 0 &0\\\ 1 & 0 & 0 &| 0 & 1 &0\\\ 0 & 0 & 1 &| 0 & 0 &1 \end{bmatrix} $$The inverse matrix can be obtained by switching the first two rows and letting the left part become a identity matrix. Then the inverse matrix is the right part:
$$ 𝐅₁⁻¹ = \begin{bmatrix} 0 & 1 &0\\\ 1 & 0 &0\\\ 0 & 0 &1 \end{bmatrix} $$It indicates that the inverse of a row-switching matrix is itself. (Just like the identity matrix, its inverse is itself.)
Inverse of a row-multiplying matrix
For a row-multiplying matrix scaling the second rows by 2 times:
$$ 𝐅₂ = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 2 & 0 \\\ 0 & 0 & 1 \end{bmatrix} $$Therefore, its inverse is obtained by multiplying 1/2 onto the 2nd row of the identity matrix.
$$ 𝐅₂⁻¹ = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1/2 & 0 \\\ 0 & 0 & 1 \end{bmatrix} $$That means the inverse of a row-multiplying matrix will still be a row-multiplying matrix. But the scale factor becomes the reciprocal.
Inverse of a row-addition matrix
For a row-addition matrix: the second row is added by 2-times first row.
$$ 𝐅₃ = \begin{bmatrix} 1 & 0 & 0 \\\ 2 & 1 & 0 \\\ 0 & 0 & 1 \end{bmatrix} $$Its inverse is obtained by lettingthe 2nd row in the identity matrix add the -2-times of the first row:
$$ 𝐅₃⁻¹ = \begin{bmatrix} 1 & 0 & 0 \\\ -2 & 1 & 0 \\\ 0 & 0 & 1 \end{bmatrix} $$Thus, its inverse is just changing the scale factor to its opposite.
In summary, the inverse matrices of elementry matrices are the same type.
- The inverse of a row-switching matrix is itself.
- The inverse of a row-multiplication matrix is changing the scale factor to its reciprocal.
- The inverse of a row-addition matrix is changing the scale factor to its opposite.
Inverse of second-order matrix
The number of rows and columns of a second-order matrix are both 2.