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# matrix that does not have a determinant

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Do all matrices have a determinant? Why or why not? Provide an example of a matrix that does not have a determinant.

Do all matrices have an inverse? Why or why not? Provide an example of a matrix that does not have an inverse.

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Check out:
http://mathworld.wolfram.com/MatrixInverse.html
http://en.wikipedia.org/wiki/Determinant

Do all matrices have a determinant? Why or why not? Provide an example of a matrix that does not have a determinant.

Only square matrices (number of columns is equal number of rows) have determinants. Since matrices in general do not have to be square, not all matrices have determinant.

For example, these matrices do not have determinant:

Do all matrices have an inverse? Why or why not? Provide an example of a matrix that does not have an inverse.

The inverse matrix of A, denoted as A-1 is defined as a matrix that satisfies:

Where is the identity matrix.
The identity matrix is a square matrix which all its entries are 0 except the diagonal which are "1".
For example a identity matrix is

We see from the definition of the inverse matrix that the matrix A and A-1 must be square and of the same order (otherwise one of the multiplications is not defined).

So any matrix which is not square does not have an inverse.
However, not even all square matrices have inverses.

Examine for example the system of linear equations:

Where x is the variables vector and b is the solution vector.

For example the system

Can be written as

Now, if A has an inverse we have:

And since I is the identity matrix, the solution for the system of equations is simply:

This indicates that only systems of equations that have a solution have an inverse matrix.
So if we have in the equations system two or more linearly dependent equations, the system has no solutions and its matrix has no inverse.

For example:

This system has no solutions since the third equation is just twice the first equation (they are linearly dependent). Thus, the matrix

Has no inverse.
An equivalent criterion for "Inverness" of a matrix is that only square matrices that their determinant is not zero have an inverse.
If then the matrix A is non-invertible.

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