2.3 Q-6

Exercise 2.3 — Question 6

Objective

Evaluate the determinant of a $3\times 3$ matrix using cofactor expansion.

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Key Concepts

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

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

Determinant by cofactor expansion along row 1: $|A|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

Singular matrix: A matrix $A$ is singular if $|A|=0$ (no inverse exists).

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Worked Example

Evaluate the determinant of $A=\left(\begin{array}{ccc}2& 3& 1\\ 1& 0& 2\\ 4& 1& 3\end{array}\right)$

Step 1 — Expand along Row 1: $|A|=2C_{11}+3C_{12}+1C_{13}$

Step 2 — Compute each cofactor:

$C_{11}=(+1)\left|\begin{array}{cc}0& 2\\ 1& 3\end{array}\right|=(0)(3)-(2)(1)=-2$

$C_{12}=(-1)\left|\begin{array}{cc}1& 2\\ 4& 3\end{array}\right|=-(3-8)=5$

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

Step 3 — Substitute: $|A|=2(-2)+3(5)+1(1)=-4+15+1=\boxed{12}$

Since $|A|=12\ne 0$, matrix $A$ is non-singular.

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Properties of Determinants (used in Q-6)

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