2.3 Q-5

Exercise 2.3 — Question 5

Topic: Evaluating Determinants Using Cofactors and Properties

This exercise focuses on evaluating $3\times 3$ determinants by cofactor expansion and applying properties of determinants.

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Key Concepts

Cofactor Expansion (along row $i$):

$|A|={\sum }_{j=1}^3{a}_{ij}\cdot C_{ij}$

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

Sign Pattern for a $3\times 3$ Matrix:

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

Useful Properties of Determinants:

• If two rows (or columns) are identical, $|A|=0$.
• If a row is all zeros, $|A|=0$.
• Interchanging two rows changes the sign of the determinant.
• A common factor of any row can be taken outside the determinant.

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Worked Example

Evaluate $\left|\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 0& 6\end{array}\right|$ by expanding along Row 1.

Step 1: Expand along Row 1:

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

Step 2: Compute each cofactor:

$C_{11}=(+1)\left|\begin{array}{cc}4& 5\\ 0& 6\end{array}\right|=(4)(6)-(5)(0)=24$

$C_{12}=(-1)\left|\begin{array}{cc}0& 5\\ 1& 6\end{array}\right|=-(0\cdot 6-5\cdot 1)=-(-5)=5$

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

Step 3: Substitute:

$|A|=1(24)+2(5)+3(-4)=24+10-12=22$

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Singular vs Non-Singular Matrices