2.2 Q-4

Exercise 2.2 — Question 4

Topic: Evaluating 3×3 Determinants Using Cofactors

SLO M-11-A-16: Evaluate the determinant of 3×3 matrices by using cofactors and properties of determinants.

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Key Concepts

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

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

Cofactor Expansion (along Row 1): $|A|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$ $=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

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Properties of Determinants (FBISE Syllabus)

1. Row/Column of Zeros: If any row or column consists entirely of zeros, then $|A|=0$.
2. Identical Rows/Columns: If any two rows (or columns) are identical, then $|A|=0$.
3. Scalar Multiple: If every element of a row (or column) is multiplied by a scalar $k$, the determinant is multiplied by $k$.
4. Row Interchange: Interchanging any two rows (or columns) changes the sign of the determinant.
5. Row Addition: Adding a scalar multiple of one row to another row does not change the value of the determinant.
6. Transpose: $|A^T|=|A|$.

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Worked Example

Evaluate the determinant of: $A=\left(\begin{array}{ccc}2& -1& 3\\ 0& 4& -2\\ 1& 5& 6\end{array}\right)$

Expanding along Row 1:

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

$C_{11}=(+1)\left|\begin{array}{cc}4& -2\\ 5& 6\end{array}\right|=(4)(6)-(-2)(5)=24+10=34$

$C_{12}=(-1)\left|\begin{array}{cc}0& -2\\ 1& 6\end{array}\right|=(-1)[(0)(6)-(-2)(1)]=(-1)(2)=-2$

$C_{13}=(+1)\left|\begin{array}{cc}0& 4\\ 1& 5\end{array}\right|=(0)(5)-(4)(1)=-4$

$|A|=2(34)+(-1)(-2)+3(-4)=68+2-12=\boxed{58}$

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Strategy: Choosing the Best Row/Column to Expand Along

To simplify calculation, expand along the row or column containing the most zeros — this minimises the number of non-zero cofactor terms to compute.

Example: For a matrix with zeros in column 2, expand along column 2 to reduce arithmetic.

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