Matrices revision notes

Matrices are one of the most powerful tools in mathematics, originally evolved to obtain compact and simple methods of solving systems of linear equations. They are heavily utilized in transformations, genetics, cryptography, and economics.

1. Matrix and Elements

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or entries of the matrix.

2. Types of Matrices

3. Equality of Matrices

Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are equal ($A = B$) if and only if:

1. They have the same order.
2. Each corresponding element is equal ($a_{ij} = b_{ij}$ for all $i, j$).

4. Operations on Matrices

5. Transpose of a Matrix ($A'$ or $A^T$)

The matrix obtained by interchanging the rows and columns of $A$. If $A = [a_{ij}]_{m \times n}$, then $A' = [a_{ji}]_{n \times m}$.

6. Symmetric and Skew-Symmetric Matrices

7. Invertible Matrices

If $A$ is a square matrix of order $m$, and there exists another square matrix $B$ of the same order such that $AB = BA = I$, then $B$ is the inverse of $A$ ($A^{-1}$).

8. Trace of a Matrix (Tr(A))

JEE Tip The sum of the principal diagonal elements of a square matrix. $\text{Tr}(A) = \sum_{i=1}^{n} a_{ii}$.

Units & Dimensions: Matrices themselves are dimensionless operators/arrays, but the elements within them carry the units of the physical quantities they represent (e.g., Cost in Rupees, Quantities in units).

1. Matrix Addition/Subtraction: Strictly requires both matrices to have the exact same dimensions ($m \times n$). Undefined otherwise.
2. Matrix Multiplication: $AB$ is ONLY valid if columns of $A$ = rows of $B$. If $A$ is $m \times n$ and $B$ is $k \times l$, $n$ must equal $k$.
3. Commutativity of Multiplication: $AB \neq BA$ in general. Even if $AB$ and $BA$ are both defined and of the same order, they are rarely equal. They only commute under specific conditions (e.g., diagonal matrices of the same order commute).
4. Invertibility: Only square matrices can be invertible. A square matrix $A$ is invertible if and only if its determinant is non-zero ($|A| \neq 0$).

• Cancellation Law Failure: If $AB = AC$, you cannot universally cancel $A$ to conclude $B = C$. This is only valid if $A$ is a non-singular square matrix (invertible).
• Algebraic Expansions: $(A+B)^2 \neq A^2 + 2AB + B^2$. Because matrix multiplication is non-commutative, $(A+B)^2 = A^2 + AB + BA + B^2$. It only collapses to the standard algebraic formula if $A$ and $B$ commute ($AB=BA$).
• Transpose and Inverse Sequence: Always remember the Reversal Law. The transpose or inverse of a product reverses the order of the matrices: $(ABC)' = C'B'A'$ and $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$.

1. Splitting a Square Matrix into Symmetric and Skew-Symmetric Parts

• Objective: Express square matrix $A$ as $P + Q$ where $P' = P$ and $Q' = -Q$.
• Step 1: Calculate $A'$.
• Step 2: Let $P = \frac{1}{2}(A + A')$. Show $P' = [\frac{1}{2}(A + A')]' = \frac{1}{2}(A' + (A')') = \frac{1}{2}(A' + A) = P$. Thus, $P$ is symmetric.
• Step 3: Let $Q = \frac{1}{2}(A - A')$. Show $Q' = [\frac{1}{2}(A - A')]' = \frac{1}{2}(A' - (A')') = \frac{1}{2}(A' - A) = -Q$. Thus, $Q$ is skew-symmetric.
• Step 4: Verify $P + Q = \frac{1}{2}A + \frac{1}{2}A' + \frac{1}{2}A - \frac{1}{2}A' = A$.

2. Proof of Reversal Law of Inverse: $(AB)^{-1} = B^{-1}A^{-1}$

• By definition of inverse: $(AB)(AB)^{-1} = I$
• Pre-multiply by $A^{-1}$: $A^{-1}(AB)(AB)^{-1} = A^{-1}I \implies (A^{-1}A)B(AB)^{-1} = A^{-1}$
• Simplify: $IB(AB)^{-1} = A^{-1} \implies B(AB)^{-1} = A^{-1}$
• Pre-multiply by $B^{-1}$: $B^{-1}B(AB)^{-1} = B^{-1}A^{-1} \implies I(AB)^{-1} = B^{-1}A^{-1}$
• Result: $(AB)^{-1} = B^{-1}A^{-1}$.

3. Using Principle of Mathematical Induction (PMI) for $A^n$

JEE Tip When asked to find $A^n$ or prove a formula for $A^n$:

• Step 1: Base case - prove the proposition $P(1)$ is true.
• Step 2: Assumption - assume $P(k)$ is true for some positive integer $k$.
• Step 3: Inductive step - multiply $P(k)$ by $A$ to find $A^{k+1}$. Prove that the resulting matrix matches the pattern for $P(k+1)$.
• Standard Result: If $A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$, then $A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}$ for all $n \in \mathbb{N}$.

1. Higher Powers of Matrices ($A^n$): Often solved using PMI, detecting periodic sequences ($A, A^2, A^3 \dots$ returning to $I$), or using the Binomial Theorem. JEE Tip If $A = I + B$ where $B$ is a nilpotent matrix (e.g., $B^2 = O$), then $A^n = (I + B)^n = I^n + nI^{n-1}B + \frac{n(n-1)}{2}I^{n-2}B^2 \dots = I + nB$.
2. Symmetric / Skew-Symmetric Combinations:
   • If $A$ and $B$ are symmetric, $AB + BA$ is symmetric, but $AB - BA$ is skew-symmetric.
   • $B'AB$ matches the symmetry/skew-symmetry of $A$.
3. Determinant of Skew-Symmetric Matrices: JEE Tip The determinant of a skew-symmetric matrix of odd order is ALWAYS zero.
4. Idempotent & Involutory Matrices:
   • Idempotent: $A^2 = A$. (e.g., $(I-A)^2 = I - A - A + A^2 = I - 2A + A = I - A$).
   • Involutory: $A^2 = I$.
5. Polynomial Matrix Equations: E.g., proving $A^3 - 6A^2 + 7A + 2I = O$. Often leads directly to finding $A^{-1}$ by multiplying the equation by $A^{-1}$.