Matrices: Inverse, Determinants & Linear Equations

TAKEAWAYS:

• Matrices are rectangular arrays of numbers used to simplify complex calculations in business (e.g., cost, profit, inventory).
• The determinant of a 2×2 or 3×3 matrix helps find its inverse (if it exists).
• A matrix has an inverse only if its determinant ≠ 0 (non-singular).
• Linear equations (e.g., supply-demand) can be solved using matrix methods like $A^{-1}X=B$.
• Applications include input-output models (economics) and least-squares regression (statistics).
• Always check if a matrix is invertible before solving equations.

---

1. What is a Matrix?

A matrix is a rectangular arrangement of numbers in rows and columns, enclosed in brackets. Matrices help organize data (e.g., sales records, cost tables) and simplify calculations.

Types of Matrices

• Square Matrix: Rows = Columns (e.g., 2×2, 3×3).
• Rectangular Matrix: Rows ≠ Columns (e.g., 2×3).
• Row Matrix: Only 1 row (e.g., [1 2 3]).
• Column Matrix: Only 1 column (e.g., $\left[\begin{array}{c}4\\ 5\\ 6\end{array}\right]$).
• Identity Matrix (I): Diagonal elements = 1, others = 0 (e.g., $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$).
• Zero Matrix (O): All elements = 0.

Matrix Operations

1. Addition/Subtraction:

   • Matrices must be of the same size.
   • Add/subtract element-wise.
   • Example: $$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$$

2. Scalar Multiplication:

   • Multiply every element by a constant (scalar).
   • Example: $$3\times \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}3& 6\\ 9& 12\end{array}\right]$$

3. Matrix Multiplication (Dot Product):

   • Columns of first matrix × Rows of second matrix.
   • Rule: Number of columns in first matrix = Number of rows in second matrix.
   • Example: $$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\times \left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}(1\times 5+2\times 7)& (1\times 6+2\times 8)\\ (3\times 5+4\times 7)& (3\times 6+4\times 8)\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$$

---

2. Determinant of a Matrix

The determinant is a scalar value that tells us if a matrix is invertible (non-singular) or not.

For a 2×2 Matrix

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\text{Det}(A)=|A|=ad-bc$$ Example: $$\left|\begin{array}{cc}4& 2\\ 6& 3\end{array}\right|=(4\times 3)-(2\times 6)=12-12=0$$ Since $|A|=0$, matrix $A$ does not have an inverse.

For a 3×3 Matrix

Use the rule of Sarrus or cofactor expansion: $$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right],|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$$ Example: $$\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|=1(5\times 9-6\times 8)-2(4\times 9-6\times 7)+3(4\times 8-5\times 7)$$ $$=1(45-48)-2(36-42)+3(32-35)=-3+12-9=0$$ Again, $|A|=0$ → No inverse.

---

3. Inverse of a Matrix

A matrix $A$ has an inverse $A^{-1}$ if:

1. It is a square matrix.
2. Its determinant ≠ 0 (non-singular).

Formula for 2×2 Matrix Inverse

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then: $$A^{-1}=\frac{1}{|A|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$ Example: Find the inverse of $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$.

1. Calculate $|A|=(1\times 4)-(2\times 3)=4-6=-2$.
2. Since $|A|\ne 0$, the inverse exists.
3. Apply the formula: $$A^{-1}=\frac{1}{-2}\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}-2& 1\\ 1.5& -0.5\end{array}\right]$$

Verification

Multiply $A$ by $A^{-1}$ to get the identity matrix: $$A\times A^{-1}=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\times \left[\begin{array}{cc}-2& 1\\ 1.5& -0.5\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

---

4. Solving Linear Equations Using Matrices

Linear equations can be written in matrix form as: $$AX=B$$ where:

• $A$ = Coefficient matrix
• $X$ = Variable matrix (unknowns)
• $B$ = Constant matrix

Solution: $$X=A^{-1}B$$ Example: Solve: $$\begin{gathered} 2x+y=5 \\ 3x+2y=11 \end{gathered}$$ Step 1: Write in matrix form: $$\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}5\\ 11\end{array}\right]$$ Step 2: Find $A^{-1}$: $$\begin{gathered} |A|=(2\times 2)-(1\times 3)=4-3=1 \\ {A}^{-1}=\frac{1}{1}\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]=\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right] \end{gathered}$$ Step 3: Multiply $A^{-1}$ by $B$: $$X=A^{-1}B=\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]\left[\begin{array}{c}5\\ 11\end{array}\right]=\left[\begin{array}{c}(2\times 5)+(-1\times 11)\\ (-3\times 5)+(2\times 11)\end{array}\right]=\left[\begin{array}{c}-1\\ 7\end{array}\right]$$ Solution: $x=-1$, $y=7$.

---

5. Applications in Business

---

6. Common Mistakes to Avoid

1. Assuming all matrices have inverses → Check $|A|\ne 0$.
2. Incorrect matrix multiplication → Columns × Rows, not rows × columns.
3. Forgetting scalar multiplication in inverse formula.
4. Mismatched matrix sizes in addition/multiplication.
5. Sign errors in determinant calculations.

---

Exam Tip

1. Always check the determinant before finding an inverse.
2. Show all steps in matrix multiplication (especially for 3×3).
3. Verify solutions by substituting back into original equations.
4. Practice 2×2 and 3×3 problems—NEB often tests both.
5. Memorize the inverse formula for 2×2 matrices.
6. For word problems, clearly define matrices $A$, $X$, and $B$ before solving.

---

NEB-Style Questions

Short Answer (5 marks each)

1. Find the inverse of the matrix $\left[\begin{array}{cc}3& -1\\ 2& 4\end{array}\right]$.
2. Solve using matrices: $$\begin{gathered} x+2y=5 \\ 3x-y=4 \end{gathered}$$
3. Calculate the determinant of $\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right]$.

Long Answer (10 marks)

4. A company has two products. The cost matrix (in Rs.) is: $$\left[\begin{array}{cc}10& 5\\ 8& 6\end{array}\right]$$ The profit matrix is $\left[\begin{array}{cc}20& 15\end{array}\right]$.
   • Find the profit if 3 units of Product 1 and 2 units of Product 2 are sold.
   • If the selling price matrix is $\left[\begin{array}{c}25\\ 20\end{array}\right]$, find the profit per unit.

Conceptual (5 marks)

5. Explain why a matrix with determinant zero cannot have an inverse. Give an example.

---

Practice Tip: Solve at least 10 problems on inverses and determinants daily to master this unit!