Verify an inverse matrix

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Verifying that a matrix has an inverse matrix is ​​obtaining the identity matrix as a consequence of multiplying the original matrix by the inverse matrix.

In other words, verifying that a matrix is ​​an inverse matrix is ​​multiplying the original matrix by the inverse matrix and obtaining the identity matrix.

Inverse matrix

An inverse matrix is ​​the linear transformation of a matrix by multiplying the inverse of the determinant of the matrix by the transposed adjoint matrix.

In other words, an inverse matrix is ​​the multiplication of the inverse of the determinant by the transposed adjoint matrix.

Property

A square matrix X of order n will have an inverse matrix X of order n, X-1, such that it fulfills the following:

Obtaining the identity matrix through the multiplication of the original matrix and the inverse matrix.

Thanks to this property we can verify that a matrix is ​​an inverse matrix.

The order of the elements of the multiplication is not relevant. That is, the multiplication of any square matrix by its inverse matrix will always result in the identity matrix of the same order.

The order of the inverse matrix is ​​the same as the order of the original matrix.

Exercise

Check that the matrix F has an inverse matrix and is the matrix U:

Square matrices

In other words, it is asked to demonstrate mathematically that

The matrix U is the inverse matrix of the matrix F

And how is that done?

If multiplying the matrix U by the matrix F we obtain the identity matrix, then, it means that the matrix U is the inverse matrix of the matrix F.

The identity matrix would be such that:

3 × 3 dimension identity matrix

Then,

Multiplication of the original matrix by the inverse matrix to obtain the identity matrix

If this equality is fulfilled, the matrix F has an inverse matrix and is the matrix U.

Transposed matrix

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