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A singular matrix is a matrix that is not invertible. If a matrix is not invertible, then:
• The determinant of the matrix is 0.
• Any matrix multiplied by that matrix doesn't give the identity matrix.
There are a lot of examples in which a singular matrix is an idempotent matrix. For instance:
M =
[1 1]
[0 0]
Take the product of two M's to get the same M, the given!
M x M = M
So yes, SOME singular matrices are idempotent matrices! How? Let's take a 2 by 2 identity matrix for instance.
I =
[1 0]
[0 1]
I x I = I obviously.
Then, that nonsingular matrix is also idempotent!
Hope this helps!
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What is a non-singuar matrix?
A non-singular matrix is basically one that has a multiplicative inverse. More specifically, a matrix "A" is non-singular if there is a matrix "B", such that AB = BA = 1, where "1" is the unity matrix. Non-singular matrixes are those that have a non-zero determinant. Singular and non-singular matrixes are only defined for square matrixes.
Why singular matrix should have non zero vector?
There is no reason why it should! So the question is based on an incorrect assumption. A matrix of only zero vectors will be singular!
When will the system of linear equation be consistent?
When its matrix is non-singular.
Is there exist a matrix whose eigenvalues are different that of its transpose?
No. Say your matrix is called A, then a number e is an eigenvalue of A exactly when A-eI is singular, where I is the identity matrix of the same dimensions as A. A-eI is singular exactly when (A-eI)T is singular, but (A-eI)T=AT-(eI)T =AT-eI. Therefore we can conclude that e is an eigenvalue of A exactly when it is an eigenvalue of AT.
What is the answer do this matrix in arbitrary form 3x-6y-z equals 6 and x plus y plus 3x equals 5?
It is a singular matrix.
