2.6 Q-9

Exercise 2.6 — Question 9

This question involves solving a system of $3\times 3$ non-homogeneous linear equations using the matrix inversion method and/or Cramer's Rule.

---

Matrix Inversion Method

For a system $AX=B$, if $|A|\ne 0$, the unique solution is:

$X=A^{-1}B$

Steps:

1. Write the system in matrix form $AX=B$.
2. Find $|A|$ (determinant of $A$) using cofactor expansion.
3. If $|A|\ne 0$, find $A^{-1}$ using row operations on the augmented matrix $[A|I]$.
4. Multiply: $X=A^{-1}B$ to get the solution.

---

Cramer's Rule

For the system: $a_{1}x+b_{1}y+c_{1}z=d_1$ $a_{2}x+b_{2}y+c_{2}z=d_2$ $a_{3}x+b_{3}y+c_{3}z=d_3$

Let $D=|A|$. Then:

$x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D}$

where:

• $D_x$ = determinant of $A$ with the first column replaced by $B$
• $D_y$ = determinant of $A$ with the second column replaced by $B$
• $D_z$ = determinant of $A$ with the third column replaced by $B$

Condition: Cramer's Rule applies only when $D=|A|\ne 0$.

---

Finding the Inverse Using Row Operations

To find $A^{-1}$, form the augmented matrix $[A|I_3]$ and apply elementary row operations until the left side becomes $I_3$. The right side then becomes $A^{-1}$.

Key row operations:

• $R_i\leftrightarrow R_j$ (swap rows)
• $R_i\to k{R}_i$ (multiply row by scalar $k\ne 0$)
• $R_i\to R_i+k{R}_j$ (add multiple of one row to another)

---

Evaluating a 3×3 Determinant by Cofactor Expansion

For matrix $A=\left(\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right)$, expanding along Row 1:

$|A|=a_1{C}_{11}+b_1{C}_{12}+c_1{C}_{13}$

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