Matrices and Determinants: Types, Operations, and Applications

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## **What is a Matrix?**
A **matrix** is a rectangular arrangement of numbers (or elements) in rows and columns, enclosed in brackets. Matrices are used to represent data, solve systems of equations, and model real-world problems.

### **Types of Matrices**
1. **Row Matrix**: Only one row.
   figure
{"type":"array","values":[[6, 8], [10, 12]],"caption":"A + B = [[6, 8], [10, 12]]"}

2. Scalar Multiplication

Multiply every element of the matrix by a scalar (a single number).

Example: If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $k=3$, then $3A=\left[\begin{array}{cc}3\times 1& 3\times 2\\ 3\times 3& 3\times 4\end{array}\right]=\left[\begin{array}{cc}3& 6\\ 9& 12\end{array}\right]$.

3. Matrix Multiplication

• Condition: Number of columns in A = Number of rows in B.
• Rule: Multiply rows of A by columns of B and sum.

Example: If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$, then $AB=\left[\begin{array}{cc}(1\times 5+2\times 7)& (1\times 6+2\times 8)\\ (3\times 5+4\times 7)& (3\times 6+4\times 8)\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$.

Note: Matrix multiplication is not commutative ($AB\ne BA$ in general).

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Transpose of a Matrix

The transpose of a matrix $A$ (denoted $A^T$) is obtained by interchanging rows and columns.

Example: If $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$, then $A^T=\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$.

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Determinant of a Matrix

The determinant is a scalar value defined only for square matrices. It helps in solving linear equations and finding inverses.

1. Determinant of a 2×2 Matrix

For $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, $$\text{det}(A)=ad-bc$$

Example: If $A=\left[\begin{array}{cc}4& 2\\ 1& 3\end{array}\right]$, $$\text{det}(A)=(4\times 3)-(2\times 1)=12-2=10$$

2. Determinant of a 3×3 Matrix (Rule of Sarrus)

For $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$, $$\text{det}(A)=a(ei-fh)-b(di-fg)+c(dh-eg)$$

Example: If $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$, $$\text{det}(A)=1(5\times 9-6\times 8)-2(4\times 9-6\times 7)+3(4\times 8-5\times 7)$$ $$=1(45-48)-2(36-42)+3(32-35)$$ $$=1(-3)-2(-6)+3(-3)$$ $$=-3+12-9=0$$

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Applications of Matrices and Determinants

1. Solving Linear Equations: Matrices help solve systems of equations using Cramer’s Rule (if det ≠ 0). Example: $$\{\begin{array}{l}2x+y=5\\ x-y=1\end{array}$$ Solution: $$x=\frac{\text{det}(A_x)}{\text{det}(A)},y=\frac{\text{det}(A_y)}{\text{det}(A)}$$ where $A_x$ replaces the first column with constants, and $A_y$ replaces the second column.

2. Geometry:

   • Rotation: Matrices rotate points in 2D/3D space.
   • Transformation: Used in computer graphics.

3. Economics:

   • Input-Output Models (Leontief Model).
   • Cost and Revenue Analysis.

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Exam Tip

1. Matrix Operations:
   • Always check if matrices are of the same order for addition/subtraction.
   • For multiplication, verify column-row compatibility.
2. Determinants:
   • For 2×2, use $ad-bc$.
   • For 3×3, use Sarrus’ rule or expansion by minors.
3. Common Mistakes:
   • Forgetting that matrix multiplication is not commutative.
   • Misapplying the determinant formula (especially signs in 3×3).
4. NEB Exam Focus:
   • Solving linear equations using matrices.
   • Finding inverses (if det ≠ 0).
   • Applications in geometry and real-world problems.

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NEB Board-Style Questions

Short Answer (5 marks each)

1. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$, find:

   • $A+B$
   • $AB$
   • $\text{det}(A)$

2. Find the transpose of $C=\left[\begin{array}{ccc}2& -1& 3\\ 0& 1& 4\end{array}\right]$.

Long Answer (10 marks)

3. Solve the system using Cramer’s Rule: $$\{\begin{array}{l}3x+2y=8\\ x-y=1\end{array}$$

4. If $D=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 4& 5& 6\end{array}\right]$, find $\text{det}(D)$.

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Key Takeaway: Matrices and determinants are powerful tools in mathematics and real-world applications. Practice operations and determinants regularly to master this unit!