Section 3.1 The Cofactor Expansion

Question: We know the determinant of a $2\times 2$ matrix. How about the determinant of an $n\times n$ matrix?

We know that if a $2\times 2$ matrix is invertible if and only if det$A\ne 0$. If we want to define the determinant of an $n\times n$ matrix then we would want the similar property as the $2\times 2$ matrix case.

Recall that a $2\times 2$ matrix is invertible if and only if it is row equivalent to the $2\times 2$ identity matrix. Here is how we operate the row reduction on a $2\times 2$ matrix:

 

$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\sim \left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{11}{a}_{21}& a_{11}{a}_{22}\end{array}\right]\sim \left[\begin{array}{cc}{a}_{11}& a_{12}\\ & a_{11}{a}_{22}-a_{12}{a}_{21}\end{array}\right]$

 

When $A$ is equivalent to identity matrix, we must have det$A$ is nonzero. Similarly, we can have the row operation on an $3\times 3$ matrix:

 

$$A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\sim \left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{11}{a}_{21}& a_{11}{a}_{22}& a_{11}{a}_{23}\\ a_{11}{a}_{31}& a_{11}{a}_{32}& a_{11}{a}_{33}\end{array}\right]\sim \left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 0& a_{11}{a}_{22}–a_{21}{a}_{12}& a_{11}{a}_{23}–a_{21}{a}_{13}\\ 0& a_{11}{a}_{32}–a_{31}{a}_{12}& a_{11}{a}_{33}–a_{31}{a}_{13}\end{array}\right]\sim \left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 0& a_{11}{a}_{22}–a_{21}{a}_{12}& a_{11}{a}_{23}–a_{21}{a}_{13}\\ 0& 0& a_{11}\mathrm{△}\end{array}\right]$$

 

where $\mathrm{△}=a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{21}{a}_{32}-a_{11}{a}_{23}{a}_{32}-a_{12}{a}_{21}{a}_{33}-a_{13}{a}_{22}{a}_{31}$. If $A$ is equivalent to the $3\times 3$ identity matrix then $\mathrm{△}$ must be nonzero. It is naturally to define the determinant of the $3\times 3$ matrix $A$ as $\mathrm{△}$. But how about $4\times 4$ matrix?

Notice that

 

$\mathrm{△}=a_{11}$det$\left[\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right]-a_{12}$det$\left[\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right]+a_{13}$det$\left[\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right]=a_{11}$det$A_{11}-a_{12}$det$A_{12}+a_{13}$det$A_{13}$

 

where $A_{11}$ is coming from deleting first row and first column of $A$ or the row and the column of the $a_{11},A_{12}$ is coming from deleting first row and the second column of $A$ or the row and the column of a_{12}, and $A_{13}$ is coming from deleting first row and third column of $A$ or the row and the column of $a_{13}$.

 

 

We use $\mathrm{△}=a_{11}$det$A_{11}-a_{12}$det$A_{12}+a_{13}$det$A_{13}$ to extend to the determinant of $n\times n$ matrix $A$.

 

Definition: Let $n\ge 2$, the determinant of an $n\times n$ matrix $A=\left[\begin{array}{c}{a}_{ij}\end{array}\right]$ is the following

 

det$A=a_{11}$det$A_{11}-a_{12}$det$A_{12}+a_{13}$det$A_{13}-a_{14}$det$A_{14}+\cdots +(-1{)}^{1+n}{a}_{1n}$det$A_{1n}=\sum _n^{j=1}(-1{)}^{1j}$det$A_{1j}$

 

where $A_{1j}$ is the submatrix of $A$ by deleting the first row and $j$-th column of $A$ or the row and the column of $a_{1j}$.

 

 

Example 1: Compute the determinant of $A=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ 2& 4& -2\end{array}\right]$.

 

 

Exercise 1: Compute the determinant of $A=\left[\begin{array}{ccc}0& 2& -2\\ 1& 0& 5\\ -1& 4& -2\end{array}\right]$.

 

Definition: Given $A=\left[\begin{array}{c}{a}_{ij}\end{array}\right]$, the ($i,j$)-cofactor of $A$ is the number $C_{ij}=(-1{)}^{i+j}$det$A_{ij}$ where $A_{ij}$ is the submtrix of $A$ by deleting $i$-th row and $j$-th column of $A$. Then

 

det$A=a_{11}{C}_{11}+a_{12}{C}_{12}+\cdots +a_{1n}{C}_{1n}$

 

is called a cofactor expansion across the first row of $A$.

 

Theorem: The determinant of an $n\times n$ matrix $A$ can be computed by a cofactor expansion across any row or down any column. The expansion across the $i$-th row is the following:

det$A=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\cdots +a_{in}{C}_{in}$

The cofactor expansion down the $j$-th column is

det$A=a_{1j}{C}_{1j}+a_{2j}{C}_{2j}+\cdots +a_{nj}{C}_{nj}$

 

Example 2: Use cofactor expansion across third row to compute det$A$ and then use the cofactor expansion across second column to compute det$A$, where $A=\left[\begin{array}{ccc}1& 0& -1\\ 0& 2& 3\\ 3& 4& -2\end{array}\right]$.

 

 

Exercise 2: Use cofactor expansion across third row to compute det$A$ and then use the cofactor expansion across second column to compute det$A$, where $A=\left[\begin{array}{ccc}2& -1& 1\\ 1& 0& 3\\ 0& 4& -2\end{array}\right]$.

 

Example 3: Compute determinant $A=\left[\begin{array}{ccccc}2& 1& -2& 1& 4\\ -1& 0& 1& -1& 2\\ 0& 0& 3& 0& 0\\ 1& 0& 2& -3& 1\\ 2& 0& 1& 3& 0\end{array}\right]$.

 

 

Exercise 3: Compute determinant $A=\left[\begin{array}{ccccc}2& 0& -2& 1& 0\\ -1& 0& 1& -1& 2\\ 3& 0& 0& 0& 0\\ 1& 1& 2& -3& -2\\ 2& 0& 1& 3& 1\end{array}\right]$.

 

Definition: A matrix is called a triangular matrix if $A$ has either upper diagonal or lower diagonal entries that are nonzero.

 

Theorem: If $A$ is a triangular matrix then det$A$ is the product of the entries on the main diagonal.

 

Fact: For an $3\times 3$ matrix $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, one can compute the determinant as the following: 1: write the first two columns of $A$ after the third column of $A$ then the determinant of $A$ is the sum of products of diagonal minus the sum of products of anti-diagonal.

 

Example 4: Compute determinant $A$ using the fact above. $A=\left[\begin{array}{ccc}1& 2& 3\\ -2& -3& 4\\ 0& 3& 5\end{array}\right]$.

 

 

Exercise 4: Compute determinant $A$ using the fact above. $A=\left[\begin{array}{ccc}0& 1& 3\\ -2& -3& 3\\ 1& 2& 4\end{array}\right]$.

 

GroupWork 1: Mark each statement True or False. Justify each answer. All matrices are $n\times n$ matrices.

a. The determinant of an $n\times n$ matrix is defined by the $(n-1)\times (n-1)$ submatrices.

 

b. The $(i,j)$-cofactor of a matrix $A=\left[\begin{array}{c}{a}_{ij}\end{array}\right]$ is the matrix $A_{ij}$ obtained by deleting from $A$ its $i$-th column and $j$-th row.

 

c. det$(2A)=2$ det$A$ when $A$ is a $2\times 2$ matrix.

 

d. If $A$ has a row or column consisting of zeros then det$A=0$.

 

e. The cofactor expansion of det$A$ down a column is the negative of the cofactor down a row.

 

f. The determinant of a triangular matrix is the sum of the diagonal matrix.

 

g. det$(-A)$ = det$A$.

 

GroupWork 2: Compute the determinant.

$E_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ k& 0& 1\end{array}\right]$

$E_2=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$

$E_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& c& 0\\ 0& 0& 1\end{array}\right]$

 

GroupWork 3: Compute det$AB$, det$A$ and det$B$. Find a relationship between them.

$A=\left[\begin{array}{ccc}1& 3& 0\\ 1& 2& -2\\ 0& 0& 1\end{array}\right]$

$B=\left[\begin{array}{ccc}-2& 5& -2\\ 1& 0& 4\\ 3& 0& 1\end{array}\right]$

 

GroupWork 4: n each case either prove the statement or give an example showing that it is false. All matrices are $n\times n$ matrices.

a. det$(A+B)$ = det$A$ + det$B$.

 

b. If det$A=0$, then $A$ has two equal rows.

 

c. If $R$ is the reduced row-echelon form of $A$, then det$A$ = det$R$.

 

d. If det$A$= det$B$ then $A=B$.

 

GroupWork 5: Find the relationship between det$A$ and det$cA$ where $c$ is a scalar.

 

GroupWork 6: If $B$ is obtained by multiply a number $c$ to one row of the matrix $A$. Find the relationship between det$A$ and det$B$.

 

GroupWork 7: If $B$ is obtained by switching two rows of the matrix $A$, find the relationship between det$A$ and det$B$.