2.2 Q-11

Exercise 2.2 — Question 11: Determinants Using Cofactors

Key Concept: Cofactor Expansion

The determinant of a $3\times 3$ matrix $A$ can be evaluated by expanding along any row or column using cofactors.

For matrix: $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Expanding along Row 1: $|A|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

where the cofactor $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ and $M_{ij}$ is the minor obtained by deleting row $i$ and column $j$.

Sign Pattern for Cofactors

$\left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right]$

Worked Example

Evaluate the determinant of: $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

Step 1: Expand along Row 1.

$|A|=1\cdot C_{11}+2\cdot C_{12}+3\cdot C_{13}$

Step 2: Compute each cofactor.

$C_{11}=(+1)\left|\begin{array}{cc}5& 6\\ 8& 9\end{array}\right|=(45-48)=-3$

$C_{12}=(-1)\left|\begin{array}{cc}4& 6\\ 7& 9\end{array}\right|=-(36-42)=6$

$C_{13}=(+1)\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|=(32-35)=-3$

Step 3: Substitute.

$|A|=1(-3)+2(6)+3(-3)=-3+12-9=0$

Properties of Determinants Used in Evaluation