2.3 Q-2

Exercise 2.3 — Q2: Evaluating 3×3 Determinants Using Cofactors

Key Concepts

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

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

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Cofactor Expansion (Laplace Expansion)

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

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

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Worked Example

Evaluate $\left|\begin{array}{ccc}2& 3& 1\\ 4& 1& 2\\ 0& 5& 3\end{array}\right|$ by expanding along Row 1.

Step 1: Find the minors: $M_{11}=\left|\begin{array}{cc}1& 2\\ 5& 3\end{array}\right|=(1)(3)-(2)(5)=3-10=-7$ $M_{12}=\left|\begin{array}{cc}4& 2\\ 0& 3\end{array}\right|=(4)(3)-(2)(0)=12$ $M_{13}=\left|\begin{array}{cc}4& 1\\ 0& 5\end{array}\right|=(4)(5)-(1)(0)=20$

Step 2: Apply cofactor signs and expand: $|A|=2(+1)(-7)+3(-1)(12)+1(+1)(20)$ $=-14-36+20=-30$

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Properties of Determinants

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Singular vs Non-Singular Matrices

• A matrix $A$ is singular if $|A|=0$ (no inverse exists).
• A matrix $A$ is non-singular if $|A|\ne 0$ (inverse exists).