2.2 Q-8

Exercise 2.2 — Question 8

Objective

Evaluate the determinant of a $3\times 3$ matrix using cofactor expansion (also called Laplace expansion) along a chosen row or column.

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Key Concepts

Cofactor of element $a_{ij}$: $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ where $M_{ij}$ is the minor — the determinant of the $2\times 2$ sub-matrix obtained by deleting row $i$ and column $j$.

Sign pattern for a $3\times 3$ matrix: $\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$

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

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Worked Example

Evaluate: $\left|\begin{array}{ccc}2& 3& -1\\ 1& 0& 2\\ 4& -2& 1\end{array}\right|$

Step 1 — Expand along Row 1:

$=2\cdot C_{11}+3\cdot C_{12}+(-1)\cdot C_{13}$

Step 2 — Compute each cofactor:

$C_{11}=(+1)\left|\begin{array}{cc}0& 2\\ -2& 1\end{array}\right|=(0)(1)-(2)(-2)=4$

$C_{12}=(-1)\left|\begin{array}{cc}1& 2\\ 4& 1\end{array}\right|=-(1\cdot 1-2\cdot 4)=-(-7)=7$

$C_{13}=(+1)\left|\begin{array}{cc}1& 0\\ 4& -2\end{array}\right|=(1)(-2)-(0)(4)=-2$

Step 3 — Substitute: $|A|=2(4)+3(7)+(-1)(-2)=8+21+2=31$

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Useful Properties of Determinants

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