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The inverse of a rotation matrix represents a rotation in the opposite direction, by the same angle, about the same axis.

Since M-1M = I, M-1(Mv) = v. Thus, any matrix inverse will "undo" the transformation of the original matrix.

Q: What does rotation mean in termsof inverse of orthogonal matrix?

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From Wolfram MathWorld: The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A-1 such that AA-1=I where I is the identity matrix.

The fact that the matrix does not have an inverse does not necessarily mean that none of the variables can be found.

A rectangular (non-square) matrix.

You can factorize the matrix using LU or LDLT factorization algorithm. inverse of a diagonal matrix (D) is really simple. To find the inverse of L, which is a lower triangular matrix, you can find the answer in this link.www.mcs.csueastbay.edu/~malek/TeX/Triangle.pdfSince (A T )-1 = (A-1 )T for all matrix, you'll just have to find inverse of L and D.

I could do that if you gave me the original matrix.

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A matrix A is orthogonal if itstranspose is equal to it inverse. So AT is the transpose of A and A-1 is the inverse. We have AT=A-1 So we have : AAT= I, the identity matrix Since it is MUCH easier to find a transpose than an inverse, these matrices are easy to compute with. Furthermore, rotation matrices are orthogonal. The inverse of an orthogonal matrix is also orthogonal which can be easily proved directly from the definition.

First let's be clear on the definitions.A matrix M is orthogonal if MT=M-1Or multiply both sides by M and you have1) M MT=Ior2) MTM=IWhere I is the identity matrix.So our definition tells us a matrix is orthogonal if its transpose equals its inverse or if the product ( left or right) of the the matrix and its transpose is the identity.Now we want to show why the inverse of an orthogonal matrix is also orthogonal.Let A be orthogonal. We are assuming it is square since it has an inverse.Now we want to show that A-1 is orthogonal.We need to show that the inverse is equal to the transpose.Since A is orthogonal, A=ATLet's multiply both sides by A-1A-1 A= A-1 ATOr A-1 AT =ICompare this to the definition above in 1) (M MT=I)do you see how A-1 now fits the definition of orthogonal?Or course we could have multiplied on the left and then we would have arrived at 2) above.

(I-A)-1 is the Leontief inverse matrix of matrix A (nxn; non-singular).

Let A by an nxn non-singular matrix, then A-1 is the inverse of A. Now (A-1 )-1 =A So the answer is yes.

If an identity matrix is the answer to a problem under matrix multiplication, then each of the two matrices is an inverse matrix of the other.

For the matrix , verify that

From Wolfram MathWorld: The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A-1 such that AA-1=I where I is the identity matrix.

No. A square matrix has an inverse if and only if its determinant is nonzero.

it is used to find the inverse of the matrix. inverse(A)= (adj A)/ mod det A

The fact that the matrix does not have an inverse does not necessarily mean that none of the variables can be found.

It need not be, so the question makes no sense!

That is called an inverse matrix