2.2 Q-13

Exercise 2.2 — Question 13

Topic: Evaluating the determinant of a $3\times 3$ matrix using cofactors.

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Key Concepts

Cofactor Expansion: For a $3\times 3$ matrix $A$, the determinant can be expanded along any row or column using cofactors.

For expansion along the first row:

$|A|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

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:

$\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$

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Worked Example (Q 13)

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 the first row.

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

Step 2: Compute each cofactor.

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

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

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

Step 3: Substitute back.

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

$\boxed{|A|=0}$

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Properties Used