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I have been learning about determinants recently and I am having difficulties understanding these three things. I completely understand them computationally, my trouble comes when I'm trying to figure out why these things where made up. I would like someone to shed light on the 'why' behind this or maybe give me a little history behind how these things came into existence.

Clemens Bartholdy
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Minors of matrices: determinants of sub-matrices that are obtained after deleting an equal number of rows and columns. Application: Laplace formula for determinants. (Sometimes the submatrices itself are called the minors.)

Adjoints of matrices: $A^T$ or $A^H$ for real or complex matrices with Euclidean inner product on $\mathbb R^n$ or $\mathbb C^n$.

Adjugates: the matrix build up of minors of $A$, which satisfies the curious identity $A \cdot \text{adj}(A) = \det(A)\cdot I$.

red_trumpet
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daw
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    The minors are not the submatrices, but rather their determinants. (I made that mistake many times myself.) – darij grinberg Jul 16 '19 at 12:05
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    @darijgrinberg reversed it, to have the 'more right' definition first. I definitely have seen both uses. – daw Jul 16 '19 at 12:10
  • @darijgrinberg, I second daw's comment that both uses circulate the literature. And recently I think more and more authors have been using "minors" to mean the submatrices themselves. – Bananeen Jul 16 '19 at 12:46
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After more than three years, I think I understand this well enough to answer. So suppose we have a $3x3$ determinant, we can think of it as a scalar product of row vectors or three column vectors. I will take the column vector picture.

In the column vector picture, the minors relate to the projection of the area along one of the planes spanned by basis. Let me make this concrete:

$$ \begin{bmatrix} p &q & r \end{bmatrix}= \begin{bmatrix} p_1 & q_1 & r_1 \\ p_2 & q_2 & r_2 \\ p_3 & q_3 & r_3 \end{bmatrix}$$

In the above, suppose we evaluate the minor $A_{22}$ , we'd have:

$$ \begin{vmatrix} p_1 &r_1 \\ p_3 & r_3\end{vmatrix}$$

What does the above mean? Well you take the first and third column vector, you find the area spanned by them, then you project that area to the x-z plane and you take the absolute value of it.

The cofactor matrix is when you don't take the absolute value of the above. We have the following relation between cofactors and minors:

$$ C_{ij} = |A_{ij}|$$

In the book I was referring in 2019, adjugate= adjoint as remarked by wiki:

The adjugate has sometimes been called the "adjoint"

I will use the word adjugate from now on. The adjugate matrix can be constructed the following way:

$$ adj( \begin{bmatrix} p &q & r \end{bmatrix}) = \begin{bmatrix} q \times r & p \times r & p \times q \end{bmatrix}$$

Where $\times$ is the cross product operator. It is clear that if we multiply ,

$$\begin{bmatrix} q \times r & p \times r & p \times q \end{bmatrix} \begin{bmatrix} p & q &r \end{bmatrix} = \det A$$

If one doesn't see the above immediately, expand everything out in components and do it. The above construction makes everything clear, and in my personal opinion, also easier to compute.

Clemens Bartholdy
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