# Definition of inverse matrix and matrix of minors

Explain what it means that a matrix $A$ is invertible. Define the inverse matrix $A^{-1}$.

I said:

A matrix $A$ is invertible if $\det(A) \neq 0$. The inverse matrix $A^{-1}$ is a matrix such that $A^{-1} \cdot A = A \cdot A^{-1} = I$ where $I$ is the identity matrix.

Would you agree with this?

Define the Minor matrix $M_{ij}$ for a square matrix $||a_{ij}||_{1 \leq i,j \leq n}$.

I said that the Minor matrix corresponding to element $i,j$ is the matrix you get after cancelling out row $i$ and column $j$. However my friend said that after you cancel out the row and column and then take the determinant, you end up with the minor matrix.

Which definition is correct?

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If you cancel row $i$ and coulmn $j$ and take the determinant you get the minor $M_{i,j}$. But this is just a number. The matrix you are looking for is probably the adjugate matrix en.wikipedia.org/wiki/Adjugate_matrix –  Henrik Finsberg May 4 '13 at 13:28