---
tag: Q47MWN8W57
kind: proof
name: Proof of the product of a matrix and its adjugate by Laplace expansion
msc: [15A09, 15A15]
proves: 0528HKWZ1Z
uses: [0GMPYM0NSR, 0W2QAVJQ1W, 076JGP6R3T, 0C08BQ3M05, 06AMCJMMVM, 0K4VQB7HWR, 0ZZQ2E1MTE, 06DW9BAWB1]
sources:
  - url: https://en.wikipedia.org/wiki/Adjugate_matrix
    locator: Important consequence
---

## Proof

We compute each entry of the two products by a Laplace expansion.

1. Let $A = (a_{kl})$ be an $n \times n$ matrix over $K$ with $n \ge 2$, and let $C_{kl}$ be its [[cofactors|0W2QAVJQ1W]].
2. By the definition of the [[adjugate|0GMPYM0NSR]], $\operatorname{adj}(A)_{lk} = C_{kl}$.
3. By the definition of the [[matrix product|076JGP6R3T]], $(A \operatorname{adj}(A))_{ik} = \sum_{j=1}^{n} a_{ij} C_{kj}$ for all $i, k$.
4. If $i = k$, the sum in step 3 is $\det A$ by the [[Laplace expansion along row $i$|0C08BQ3M05]].
5. Let $i \ne k$. Let $A'$ be the matrix that results when row $k$ of $A$ is replaced by row $i$ of $A$.
6. The cofactor $C'_{kj}$ of $A'$ deletes row $k$, the only row where $A'$ and $A$ differ. So $C'_{kj} = C_{kj}$ for all $j$.
7. Row $k$ of $A'$ has entries $a_{ij}$. By the Laplace expansion of $A'$ along row $k$ and step 6, $\det A' = \sum_{j=1}^{n} a_{ij} C_{kj}$.
8. Rows $i$ and $k$ of $A'$ are equal. By the rule for [[two equal rows|0K4VQB7HWR]], $\det A' = 0$.
9. By steps 3 to 8, $(A \operatorname{adj}(A))_{ik}$ is $\det A$ if $i = k$ and $0$ if $i \ne k$. So $A \operatorname{adj}(A) = \det(A)\, I_n$, with the [[identity matrix|06DW9BAWB1]] $I_n$.
10. In the same way, $(\operatorname{adj}(A) A)_{ik} = \sum_{j=1}^{n} C_{ji} a_{jk}$ for all $i, k$.
11. If $i = k$, this sum is $\det A$ by the [[Laplace expansion along column $i$|06AMCJMMVM]].
12. Let $i \ne k$. Let $A''$ be the matrix that results when column $i$ of $A$ is replaced by column $k$ of $A$.
13. The cofactor $C''_{ji}$ of $A''$ deletes column $i$, the only column where $A''$ and $A$ differ. So $C''_{ji} = C_{ji}$ for all $j$.
14. Column $i$ of $A''$ has entries $a_{jk}$. By the Laplace expansion of $A''$ along column $i$, $\det A'' = \sum_{j=1}^{n} a_{jk} C_{ji}$.
15. Columns $i$ and $k$ of $A''$ are equal, so the transpose of $A''$ has two equal rows. By the rule for the [[determinant of a transpose|0ZZQ2E1MTE]] and step 8's rule, $\det A'' = 0$.
16. By steps 10 to 15, $\operatorname{adj}(A) A = \det(A)\, I_n$. Together with step 9, $A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A)\, I_n$.

## Idea

An entry of $A \operatorname{adj}(A)$ pairs one row of $A$ with the cofactors of another row. For the same row this is the Laplace expansion of $\det A$. For different rows it is the expansion of a matrix with two equal rows, which is $0$.