2.2 Matrix Construction and Operations

This section covers constructing matrices from formulas, performing matrix operations, and proving matrix identities — all core skills for FBISE Grade 11 Mathematics.

---

1. Constructing a Matrix from a Formula

A matrix $A=[a_{ij}]$ of order $m\times n$ can be defined by giving a formula for each element $a_{ij}$, where:

• $i$ = row number ($1\le i\le m$)
• $j$ = column number ($1\le j\le n$)

Example

Construct a $2\times 3$ matrix $B=[b_{ij}]$ where $b_{ij}=\frac{i^2-j}{3}$.

Calculate each element by substituting the appropriate $i$ and $j$:

$b_{11}=\frac{1-1}{3}=0,b_{12}=\frac{1-2}{3}=-\frac{1}{3},b_{13}=\frac{1-3}{3}=-\frac{2}{3}$

$b_{21}=\frac{4-1}{3}=1,b_{22}=\frac{4-2}{3}=\frac{2}{3},b_{23}=\frac{4-3}{3}=\frac{1}{3}$

$B=\left[\begin{array}{ccc}0& -\frac{1}{3}& -\frac{2}{3}\\ 1& \frac{2}{3}& \frac{1}{3}\end{array}\right]$

---

2. Matrix Multiplication

For matrices $A$ (of order $m\times n$) and $B$ (of order $n\times p$), the product $AB$ is an $m\times p$ matrix where:

$(AB{)}_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$

Key properties:

• Matrix multiplication is not commutative in general: $AB\ne BA$
• It is associative: $(AB)C=A(BC)$
• It is distributive: $A(B+C)=AB+AC$

Solving Matrix Equations

To solve $AXB=C$ for unknown matrix $X$ (where $A$ and $B$ are invertible):

1. Pre-multiply both sides by $A^{-1}$: $XB=A^{-1}C$
2. Post-multiply both sides by $B^{-1}$: $X=A^{-1}C{B}^{-1}$

---

3. Transpose Properties

The transpose of a matrix $A$, written $A^T$, is obtained by swapping rows and columns.

Key properties:

The product rule extends: $(ABC{)}^T=C^T{B}^T{A}^T$.

---

4. Proving Matrix Identities

Idempotent-type Identities

Given conditions like $AB=B$ and $BA=A$, we can simplify powers of matrices:

Simplifying $A^2$: $A^2=A\cdot A=A(BA)=(AB)A=B\cdot A=A$

Simplifying $B^2$: $B^2=B\cdot B=B(AB)=(BA)B=A\cdot B=B$

Therefore: $A^2+B^2=A+B$

Polynomial Identities in Matrices

To verify an identity like $X^2-4X-5I=O$ for a given matrix $X$:

1. Compute $X^2$ by matrix multiplication.
2. Compute $4X$ by scalar multiplication.
3. Note: the constant $5$ must be written as $5I$ (where $I$ is the identity matrix of the same order as $X$) to allow matrix subtraction.
4. Substitute and simplify to show the result is the zero matrix $O$.

---

5. Proving $A^n$ by Mathematical Induction

To prove a formula for $A^n$ for all positive integers $n$:

1. Base case ($n=1$): Verify the formula holds for $A^1=A$.
2. Inductive hypothesis: Assume the formula holds for $n=k$, i.e., assume $A^k=\text{[formula]}$.
3. Inductive step: Show it holds for $n=k+1$ by computing $A^{k+1}=A^k\cdot A$ and using the hypothesis.

---

Summary of Key Rules