Matrices and Linear Algebra: Systems, Determinants, and Applications

TAKEAWAYS:

• Matrices are rectangular arrays of numbers used to represent systems of equations, optimize resource allocation (e.g., inventory, budgets), and model real-world constraints (e.g., traffic flow, production limits).
• Determinants measure the scalability of a matrix and enable Cramer’s rule to solve linear systems uniquely when the determinant is non-zero.
• Gaussian elimination transforms a system into row-echelon form to reveal solutions, while the matrix method uses inverse matrices for exact solutions (if the inverse exists).
• Applications include loan amortization schedules (banks), supply chain logistics (Daraz), and traffic signal optimization (Kathmandu Municipality).
• Exam focus: Prioritize solving systems using both Cramer’s rule and matrix inversion, and recognize when a system has no solution or infinite solutions (singular matrices).
• Visual tools: Always draw augmented matrices for elimination and graph linear systems to interpret solutions geometrically.

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1. Introduction to Matrices

Matrices are fundamental tools in business mathematics for organizing data and solving systems of linear equations. They appear in inventory management, financial modeling, and network flows.

1.1 Definition and Types

A matrix is a rectangular array of numbers arranged in rows and columns. For example:

• A 2×3 matrix has 2 rows and 3 columns:
  • Order: $m\times n$ (rows × columns).
  • Elements: Denoted as $a_{ij}$, where $i$ is the row and $j$ is the column.

Types of Matrices:

1.2 Matrix Operations

1.2.1 Addition and Subtraction

• Condition: Matrices must have the same order.
• Rule: Add/subtract corresponding elements.

1.2.2 Scalar Multiplication

Multiply every element by a scalar (constant):

1.2.3 Matrix Multiplication

• Condition: Number of columns in the first matrix must equal the number of rows in the second.
• Rule: Dot product of rows and columns. Note: Matrix multiplication is not commutative ($A\times B\ne B\times A$).

1.2.4 Transpose of a Matrix

Swap rows and columns:

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2. Determinants

Determinants are scalar values associated with square matrices. They determine whether a matrix has an inverse and are used in Cramer’s rule.

2.1 Definition and Calculation

For a 2×2 matrix: $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\det (A)=ad-bc$$ For a 3×3 matrix, use the rule of Sarrus or Laplace expansion: $$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right],\det (A)=a(ei-fh)-b(di-fg)+c(dh-eg)$$

2.2 Properties of Determinants

2.3 Cramer’s Rule

Solves a system of linear equations using determinants. For a system $AX=B$, the solution for $x_i$ is: $$x_i=\frac{\det (A_i)}{\det (A)}$$ where $A_i$ is the matrix $A$ with the $i$-th column replaced by $B$.

Example: Solve $4x+6z=100$, $3x+6y+z=100$, $3x+4y+3z=100$.

1. Write the augmented matrix and compute $\det (A)$: $$A=\left[\begin{array}{ccc}4& 0& 6\\ 3& 6& 1\\ 3& 4& 3\end{array}\right],\det (A)=4(18-4)-0(9-3)+6(12-18)=4\ast 14+6\ast (-6)=56-36=20$$
2. Replace columns to find $\det (A_x)$, $\det (A_y)$, $\det (A_z)$: $$A_x=\left[\begin{array}{ccc}100& 0& 6\\ 100& 6& 1\\ 100& 4& 3\end{array}\right],\det (A_x)=100(18-4)-0+6(400-100)=1400+1800=3200$$ $$x=\frac{3200}{20}=160$$ Similarly, compute $y$ and $z$: $$y=\frac{\det (A_y)}{\det (A)}=\frac{200}{20}=10,z=\frac{\det (A_z)}{\det (A)}=\frac{400}{20}=20$$ Solution: $(x,y,z)=(160,10,20)$.

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3. Solving Systems of Linear Equations

Three methods are commonly used:

1. Matrix Method (Inverse Method)
2. Gaussian Elimination
3. Cramer’s Rule

3.1 Matrix Method

For $AX=B$, the solution is $X=A^{-1}B$, provided $\det (A)\ne 0$.

Example: Solve $x+y+z=5$, $2x-y+3z=4$, $4x+5y+10z=13$.

1. Write $A$ and $B$: $$A=\left[\begin{array}{ccc}1& 1& 1\\ 2& -1& 3\\ 4& 5& 10\end{array}\right],B=\left[\begin{array}{c}5\\ 4\\ 13\end{array}\right]$$
2. Compute $A^{-1}$:
   • Find $\det (A)=1(-10+15)-1(20-12)+1(10+4)=5-8+14=11$.
   • Use adjugate method to find $A^{-1}$.
3. Multiply $A^{-1}B$ to get $X$.

Solution: $(x,y,z)=(1,2,2)$.

3.2 Gaussian Elimination

Transform the augmented matrix $[A|B]$ into row-echelon form and solve by back-substitution.

Example: Solve $x+2y+3z=13$, $2x+4y+z=11$, $3x+2y+2z=14$.

1. Write the augmented matrix: $$\left[\begin{array}{ccccc}1& 2& 3& |& 13\\ 2& 4& 1& |& 11\\ 3& 2& 2& |& 14\end{array}\right]$$
2. Perform row operations:
   • $R_2\to R_2-2R_1$
   • $R_3\to R_3-3R_1$
3. Resulting matrix: $$\left[\begin{array}{ccccc}1& 2& 3& |& 13\\ 0& 0& -5& |& -15\\ 0& -4& -7& |& -25\end{array}\right]$$
4. Interpretation: The second row implies $0=3$, which is a contradiction. Thus, no solution exists.

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4. Applications in Business

4.1 Real-World Examples

1. eSewa and Khalti (Digital Payments)

   • Idea Used: Matrix Multiplication for Transaction Routing
   • How: When you transfer money from Khalti to eSewa, the system uses matrices to route transactions through multiple servers. Each server’s processing capacity is represented as a matrix, and the system multiplies these matrices to optimize the path with the least latency.
   • Example: Suppose Khalti has 3 servers with processing speeds $A=[2,3,1]$ (transactions/sec) and eSewa has 2 servers with speeds $B=[4,2]$. The combined routing efficiency is computed as $A\times B^T$.

2. Daraz (E-Commerce Logistics)

   • Idea Used: Systems of Equations for Inventory Management
   • How: Daraz uses matrices to balance inventory across warehouses. Suppose a product has demand equations: $$\{\begin{array}{l}x+y=1000\text{ (total stock)}\\ 2x+y=1500\text{ (sales constraint)}\end{array}$$ Solving this system ensures optimal stock distribution.

3. NTC (Telecom Network Optimization)

   • Idea Used: Determinants for Network Stability
   • How: NTC uses determinants to check if their network routing matrices are invertible (i.e., no deadlocks). If $\det (A)=0$, the network has redundant paths causing instability.

4.2 Worked Example: Kathmandu Traffic Signal Optimization

Problem: A traffic intersection has 3 roads with flow rates represented by the matrix: $$A=\left[\begin{array}{ccc}500& 300& 200\\ 400& 600& 100\\ 300& 200& 700\end{array}\right]$$ where rows represent time slots (morning, afternoon, evening) and columns represent roads. The city wants to minimize congestion by adjusting signal timings (represented by vector $X$): $$AX=B,B=\left[\begin{array}{c}1000\\ 1100\\ 1200\end{array}\right]$$ Solution:

1. Compute $A^{-1}$ and solve $X=A^{-1}B$.
2. The resulting $X$ gives optimal signal durations per road.

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5. Advantages and Disadvantages

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6. Exam Tips

1. Check Determinant First: Before solving, always compute $\det (A)$. If zero, the system may have no solution or infinite solutions.
2. Matrix Method vs. Cramer’s Rule:
   • Use Cramer’s rule for small systems (≤3 equations) where determinants are easy to compute.
   • Use matrix inversion or Gaussian elimination for larger systems.
3. Graphical Interpretation:
   • For 2 equations, plot the lines to visualize solutions (intersection point, parallel lines, or coincident lines).
4. Common Mistakes:
   • Forgetting to check matrix order for addition/multiplication.
   • Incorrectly computing determinants (sign errors in expansion).
   • Misapplying Cramer’s rule (replacing wrong columns in $A_i$).
5. Practice:
   • Memorize the rule of Sarrus for 3×3 determinants.
   • Know when to stop in Gaussian elimination (contradiction or free variables).

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