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?

  • $\begingroup$ I have a presentation coming up and these are some of the things I will need to define so I just wanted to make sure what I had was correct. $\endgroup$ – Kaish May 3 '13 at 18:59
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    $\begingroup$ Too many question in one. $\endgroup$ – lhf May 3 '13 at 19:02
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    $\begingroup$ @lhf So do I have to delete them and put them in other ones? Surely they should be answerable by one person as they all come from a similar topic? $\endgroup$ – Kaish May 3 '13 at 19:04
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    $\begingroup$ Separate the questions. It's confusing and bad for searchability otherwise. $\endgroup$ – Qiaochu Yuan May 4 '13 at 1:22

I would not use the determinant in the definition of an invertible matrix. This is a more general definition :

Let $\mathbb{K}$ be a field (for instance you can take $\mathbb{K}=\mathbb{C}$) and $I_n$ the identity matrix of size $n\times n$. ($M_n(\mathbb{K})$ denotes the set of all squared matrices of size $n\times n$ with coefficients in $\mathbb{K}$).

Definition: A matrix $A\in M_n(\mathbb{K})$ is said to be invertible if there exists a matrix $B\in M_n(\mathbb{K})$ such that $AB = I_n$ and $BA = I_n$. We write $B=A^{-1}$.

Concerning your second question, I do not think there is such thing as a minor matrix (at least I have never heard of it). From what I know, the minor $(i,j)$ is defined this way :

Definition: Let $A\in M_n(\mathbb{K})$. The minor $(i,j)$ is the determinant of the $(n-1)\times(n-1)$ matrix created by deleting the i-th row and j-th column of A.

  • $\begingroup$ Sorry I didn't answer both questions at once. I edited my answer with the definition of a minor. $\endgroup$ – user41489 May 8 '13 at 20:22

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