Matrices & Determinants – definitions, operations, inverses, and applications

1. Basic Definitions

Types of Matrices

• Row matrix: $1\times n$
• Column matrix: $m\times 1$
• Square matrix: $n\times n$
• Zero (null) matrix: all entries 0
• Diagonal matrix: non‑zero entries only on the main diagonal
• Scalar matrix: diagonal matrix with equal diagonal entries
• Identity matrix: special scalar matrix with diagonal entry 1
• Symmetric matrix: $A=A^T$

2. Matrix Arithmetic

2.1 Addition & Subtraction

Only possible when matrices have the same order.
$$C=A+B⟹c_{ij}=a_{ij}+b_{ij}$$

2.2 Scalar Multiplication

$$kA=[k{a}_{ij}]$$

2.3 Matrix Multiplication

If $A$ is $m\times p$ and $B$ is $p\times n$, the product $C=AB$ is $m\times n$ with
$$c_{ij}=\sum _{k=1}^p{a}_{ik}{b}_{kj}$$

flowchart LR
    subgraph A ["Matrix A (2x3)"]
        direction TB
        A11["1"] --- A12["2"] --- A13["0"]
        A21["-1"] --- A22["3"] --- A23["4"]
    end
    subgraph B ["Matrix B (3x2)"]
        direction TB
        B11["2"] --- B12["-1"]
        B21["0"] --- B22["5"]
        B31["3"] --- B32["1"]
    end
    subgraph C ["Product C (2x2)"]
        direction TB
        C11["2"] --- C12["9"]
        C21["10"] --- C22["20"]
    end
    A11 --> C11
    A12 --> C11
    A13 --> C11
    A21 --> C21
    A22 --> C21
    A23 --> C21
    B11 --> C11
    B21 --> C11
    B31 --> C11
    B12 --> C12
    B22 --> C12
    B32 --> C12
    B11 --> C21
    B21 --> C21
    B31 --> C21
    B12 --> C22
    B22 --> C22
    B32 --> C22

Worked Example 1 – Multiplying Two Matrices

$$A=\left[\begin{array}{ccc}1& 2& 0\\ -1& 3& 4\end{array}\right],B=\left[\begin{array}{cc}2& -1\\ 0& 5\\ 3& 1\end{array}\right]$$

Compute $C=AB$.

$$\begin{array}{rl}{c}_{11}& =1\cdot 2+2\cdot 0+0\cdot 3=2\\ c_{12}& =1\cdot (-1)+2\cdot 5+0\cdot 1=-1+10=9\\ c_{21}& =(-1)\cdot 2+3\cdot 0+4\cdot 3=-2+12=10\\ c_{22}& =(-1)\cdot (-1)+3\cdot 5+4\cdot 1=1+15+4=20\end{array}$$

$$\boxed{C=\left[\begin{array}{cc}2& 9\\ 10& 20\end{array}\right]}$$

3. Transpose and Symmetry

The transpose flips rows and columns: $(A^T{)}_{ij}=a_{ji}$.
A matrix is symmetric if $A=A^T$.

4. Determinants

Only defined for square matrices.

4.1 Determinant of a 2×2 Matrix

$$\left|\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right|=ad-bc$$

4.2 Determinant of a 3×3 Matrix (Laplace Expansion)

$$|A|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$$ where $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ and $M_{ij}$ is the minor obtained by deleting row $i$ and column $j$.

flowchart TD
    A["Matrix A"] --> B["Expand along Row 1"]
    B --> C1["Term 1: +2 * Minor(1,1)"]
    B --> C2["Term 2: -(-1) * Minor(1,2)"]
    B --> C3["Term 3: +0 * Minor(1,3)"]
    C1 --> D1["Minor(1,1) = |3 1; 5 -2| = -11"]
    C2 --> D2["Minor(1,2) = |4 1; 0 -2| = -8"]
    C3 --> D3["Minor(1,3) = |4 3; 0 5| = 20"]
    D1 --> E["2 * (-11) = -22"]
    D2 --> F["1 * (-8) = -8"]
    D3 --> G["0 * 20 = 0"]
    E --> H["Sum: -22 - 8 + 0"]
    F --> H
    G --> H
    H --> I["|A| = -30"]

Worked Example 2 – Determinant of a 3×3 Matrix

Using the matrix from the first figure:

$$\begin{array}{rl}|A|& =2\cdot \left|\begin{array}{cc}3& 1\\ 5& -2\end{array}\right|-(-1)\cdot \left|\begin{array}{cc}4& 1\\ 0& -2\end{array}\right|+0\cdot \left|\begin{array}{cc}4& 3\\ 0& 5\end{array}\right|\\ & =2(3\cdot (-2)-1\cdot 5)+1(4\cdot (-2)-1\cdot 0)+0\\ & =2(-6-5)+1(-8)\\ & =2(-11)-8=-22-8=-30\end{array}$$

$$\boxed{|A|=-30}$$

Since $|A|\ne 0$, the matrix is invertible.

5. Inverse of a Square Matrix

A matrix $A$ has an inverse $A^{-1}$ iff $|A|\ne 0$. Two common methods:

1. Adjugate method (for small orders)
   $$A^{-1}=\frac{1}{|A|}\mathrm{adj}(A)$$ where $\mathrm{adj}(A)$ is the transpose of the cofactor matrix.

2. Gauss‑Jordan elimination (row‑reduction of $[A|I]$ to $[I|A^{-1}]$).

Worked Example 3 – Inverse of a 2×2 Matrix

$$A=\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right]$$

$$|A|=4\cdot 6-7\cdot 2=24-14=10\ne 0$$

Adjugate: $$\mathrm{adj}(A)={\left[\begin{array}{cc}6& -7\\ -2& 4\end{array}\right]}^T=\left[\begin{array}{cc}6& -2\\ -7& 4\end{array}\right]$$

$$A^{-1}=\frac{1}{10}\left[\begin{array}{cc}6& -2\\ -7& 4\end{array}\right]=\left[\begin{array}{cc}0.6& -0.2\\ -0.7& 0.4\end{array}\right]$$

6. Properties of Determinants (Comparison Table)

| Property                              | Description                                          |
|---------------------------------------|------------------------------------------------------|
| \(|AB| = |A|\;|B|\)                    | Determinant of a product equals product of determinants |
| \(|A^{T}| = |A|\)                     | Transpose does not change determinant                |
| \(|kA| = k^{n}|A|\) (for \(n\times n\))| Scaling a matrix by \(k\) multiplies determinant by \(k^{n}\) |
| Swapping two rows (or columns) → sign change | \(|A'| = -|A|\) after one row swap |
| Adding a multiple of one row to another → determinant unchanged | Useful for row‑reduction |

7. Applications

8. Worked Real‑World Example

Scenario: A small Nepali shop uses a 2‑item inventory model. Let

flowchart LR
    A["Inventory Matrix A"] --> B["Check Determinant"]
    B --> C{"|A| != 0?"}
    C -- Yes --> D["Calculate Inverse A^-1"]
    C -- No --> E["No Unique Solution"]
    D --> F["Multiply A^-1 by b"]
    F --> G["Solution Vector x"]
    G --> H["x1 = 8, x2 = 6"]

$$A=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$$

where column 1 = units of Item X needed per product, column 2 = units of Item Y needed per product. The shop wants to produce a mix $\mathbf{x}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$ that uses exactly 30 units of X and 40 units of Y.

Solve $A\mathbf{x}=\mathbf{b}$ with $\mathbf{b}=\left[\begin{array}{c}30\\ 40\end{array}\right]$.

1. Compute $|A|=3\cdot 4-1\cdot 2=12-2=10$.
2. Find $A^{-1}=\frac{1}{10}\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]$.
3. Multiply: $\mathbf{x}=A^{-1}\mathbf{b}$

$$\mathbf{x}=\frac{1}{10}\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]\left[\begin{array}{c}30\\ 40\end{array}\right]=\frac{1}{10}\left[\begin{array}{c}4\cdot 30-1\cdot 40\\ -2\cdot 30+3\cdot 40\end{array}\right]=\frac{1}{10}\left[\begin{array}{c}120-40\\ -60+120\end{array}\right]=\frac{1}{10}\left[\begin{array}{c}80\\ 60\end{array}\right]=\left[\begin{array}{c}8\\ 6\end{array}\right]$$

Interpretation: Produce 8 units of product 1 and 6 units of product 2 to exhaust the stock exactly.

9. In‑Class Tips for Solving Determinants Quickly

• Row‑Reduction: Convert to upper‑triangular form; determinant = product of diagonal entries (adjust sign for row swaps).
• Sarrus’ Rule (only for 3×3): repeat first two columns, sum down‑diagonals, subtract up‑diagonals.
• Cofactor Expansion: Choose the row/column with most zeros to minimise calculations.

In the real world

• Google Search – PageRank algorithm treats the web as a directed graph; the transition matrix’s eigen‑vector (derived from the matrix) ranks pages.
• Daraz Order Management – Inventory across warehouses is represented by a matrix; multiplying the order‑quantity vector by the stock‑availability matrix yields fulfillment feasibility.
• Ncell Network Planning – The adjacency matrix of cell‑tower connections is used; its determinant of the Laplacian matrix tells engineers how many independent spanning trees exist, indicating network robustness.

Exam tip

• Determinant shortcuts: memorize Sarrus for $3\times 3$ and the 2×2 formula; practice row‑reduction to triangular form for larger matrices.
• Inverse: For $2\times 2$ always use the adjugate formula; for $3\times 3$ or higher, be comfortable with the Gauss‑Jordan method – write the augmented matrix $[A|I]$ and perform elementary row operations until the left side becomes $I$.
• Marking scheme: Full credit is given for each correct elementary operation (row swap, scaling, addition) and for stating the effect on the determinant. Skip unnecessary intermediate steps to save time.