Matrices and Determinants: Types, Operations, and Business Uses

TAKEAWAYS:

• Matrices are rectangular arrays of numbers used to organize data (e.g., inventory, costs) and solve systems of equations.
• Determinants help find unique solutions for square matrices and are used in business for profit maximization and cost analysis.
• Matrix operations (addition, subtraction, multiplication) follow strict rules—only like-sized matrices can be added/subtracted, and multiplication requires column-row matching.
• The inverse of a matrix (A⁻¹) exists only if its determinant is non-zero, and it reverses multiplication (AA⁻¹ = I).
• Cramer’s Rule uses determinants to solve linear equations, while profit optimization problems often involve matrix equations.
• NEB exams test calculations, real-world applications (e.g., cost matrices), and conceptual questions on properties.

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What is a Matrix?

A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. Matrices are used to represent data, solve systems of equations, and model real-world problems like cost, inventory, or production.

Types of Matrices

1. Row Matrix: Only one row (e.g., [5 8 12]).
2. Column Matrix: Only one column (e.g., [[3], [7], [11]]).
3. Square Matrix: Equal rows and columns (e.g., 2×2, 3×3).
4. Diagonal Matrix: Non-zero elements only on the diagonal (e.g., [[4, 0], [0, 5]]).
5. Identity Matrix (I): Diagonal elements are 1, others 0 (e.g., [[1, 0], [0, 1]]).
6. Zero Matrix (O): All elements are 0.

Matrix Operations

1. Addition and Subtraction

• Rule: Matrices must be of the same size.
• Example: Let $A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}5& -1\\ 0& 2\end{array}\right]$. Then, $A+B=\left[\begin{array}{cc}2+5& 3+(-1)\\ 1+0& 4+2\end{array}\right]=\left[\begin{array}{cc}7& 2\\ 1& 6\end{array}\right]$.

2. Scalar Multiplication

Multiply every element by a scalar (number).

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

3. Matrix Multiplication

• Rule: Number of columns in the first matrix must equal the number of rows in the second.
• Example: Let $C=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $D=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$. Then, $CD=\left[\begin{array}{cc}(1\times 5+2\times 7)& (1\times 6+2\times 8)\\ (3\times 5+4\times 7)& (3\times 6+4\times 8)\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$.

Note: Matrix multiplication is not commutative (i.e., $AB\ne BA$).

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Determinants

The determinant of a square matrix is a scalar value that determines if the matrix has an inverse and helps solve systems of equations.

Calculating Determinants

1. For a 2×2 Matrix

For $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the determinant is: $$\text{det}(A)=ad-bc$$ Example: For $A=\left[\begin{array}{cc}4& 2\\ 1& 3\end{array}\right]$, $$\text{det}(A)=(4\times 3)-(2\times 1)=12-2=10.$$

2. For a 3×3 Matrix

For $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$, the determinant is: $$\text{det}(A)=a(ei-fh)-b(di-fg)+c(dh-eg).$$ Example: For $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$, $$\text{det}(A)=1(5\times 9-6\times 8)-2(4\times 9-6\times 7)+3(4\times 8-5\times 7)$$ $$=1(45-48)-2(36-42)+3(32-35)=-3+12-9=0.$$

Properties of Determinants

1. The determinant of a triangular matrix is the product of its diagonal elements.
2. Swapping two rows/columns changes the sign of the determinant.
3. If any row/column is all zeros, the determinant is zero.
4. Multiplying a row/column by a scalar multiplies the determinant by that scalar.

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Inverse of a Matrix

The inverse of a matrix $A$ (denoted $A^{-1}$) is a matrix such that: $$A{A}^{-1}=A^{-1}A=I.$$ Condition: The inverse exists only if $\text{det}(A)\ne 0$.

Finding the Inverse of a 2×2 Matrix

For $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the inverse is: $$A^{-1}=\frac{1}{\text{det}(A)}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right].$$ Example: For $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, $$\text{det}(A)=(1\times 4)-(2\times 3)=-2.$$ $$A^{-1}=\frac{1}{-2}\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}-2& 1\\ 1.5& -0.5\end{array}\right].$$

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Applications in Business

1. Solving Systems of Equations: Use Cramer’s Rule (determinants) to solve linear equations. Example: Solve $2x+y=5$ and $x-y=1$. Write as $\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}5\\ 1\end{array}\right]$. Use determinants to find $x$ and $y$.

2. Cost and Profit Optimization: Matrices represent cost, production, and profit data. Inverses help find optimal allocations.

3. Input-Output Models in Economics: Used to analyze how changes in one sector affect others.

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Exam Tip

1. Memorize Rules: Matrix multiplication rules, determinant formulas, and inverse conditions.
2. Practice Calculations: Solve 2×2 and 3×3 determinants and inverses repeatedly.
3. Real-World Problems: Expect questions on cost matrices, profit optimization, or solving equations using matrices.
4. Avoid Common Mistakes:
   • Forgetting to check matrix sizes for addition/multiplication.
   • Misapplying the determinant formula for 3×3 matrices.
   • Assuming $AB=BA$ (matrix multiplication is not commutative).

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NEB-Style Questions

Short Answer

1. Define a square matrix. Give an example.
2. What is the determinant of $\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]$?
3. When does a matrix not have an inverse?

Long Answer

1. Solve the system using determinants: $3x+2y=8$ $4x-y=5$.

2. A company’s cost matrix for producing two products is: $\left[\begin{array}{cc}5& 3\\ 2& 4\end{array}\right]$. If the demand matrix is $\left[\begin{array}{c}10\\ 15\end{array}\right]$, find the total cost using matrix multiplication.

Objective Questions

1. The determinant of $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is: a) $ad+bc$ b) $ad-bc$ c) $ab-cd$ d) $ac-bd$

2. Which of the following is true for matrix multiplication? a) $AB=BA$ b) $(A+B)C=AC+BC$ c) $A(BC)=(AB)C$ d) $A+B=B+A$

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Answer Key:

1. b) $ad-bc$
2. c) $A(BC)=(AB)C$