Matrices revision notes

Key Concepts & Definitions

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.

Representation

Any point (x,y)(x, y)(x,y) in a plane can be represented by a column or row matrix. Vertices of closed rectilinear figures (like quadrilaterals) can also be formulated as matrices.

Order of a matrix

A matrix having mmm rows and nnn columns is of order m×nm \times nm×n (read as "mmm by nnn"). Total number of elements = mnmnmn.

General Form

A=[aij]m×nA = [a_{ij}]_{m \times n}A=[aij​]m×n​, where 1≤i≤m1 \le i \le m1≤i≤m, 1≤j≤n1 \le j \le n1≤j≤n, and i,j∈Ni, j \in \mathbb{N}i,j∈N. The element aija_{ij}aij​ lies in the ithi^{th}ith row and jthj^{th}jth column.

2. Types of Matrices

• Column Matrix: A matrix with exactly one column. Order: $m \times 1$.
• Row Matrix: A matrix with exactly one row. Order: $1 \times n$.
• Square Matrix: A matrix where the number of rows equals the number of columns ($m = n$). Order is simply denoted as $n$.
  • Diagonal Elements: For a square matrix $A = [a_{ij}]_{n \times n}$, the elements $a_{11}, a_{22}, \dots, a_{nn}$ constitute the principal diagonal.
• Diagonal Matrix: A square matrix where all non-diagonal elements are zero ($b_{ij} = 0$ when $i \neq j$).
• Scalar Matrix: A diagonal matrix where all diagonal elements are equal ($b_{ij} = 0$ for $i \neq j$; $b_{ij} = k$ for $i = j$).
• Identity (Unit) Matrix: A scalar matrix where all diagonal elements are $1$. Denoted by $I_n$ or simply $I$. ($a_{ij} = 1$ if $i = j$, $0$ if $i \neq j$).
• Zero / Null Matrix: A matrix (of any order) where all elements are zero. Denoted by $O$.
• Triangular Matrices:JEE TIP
  • Upper Triangular: Square matrix where all elements below the principal diagonal are zero ($a_{ij} = 0$ for $i > j$).
  • Lower Triangular: Square matrix where all elements above the principal diagonal are zero ($a_{ij} = 0$ for $i < j$).

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

• Addition: Defined ONLY for matrices of the same order. $C = A + B \implies c_{ij} = a_{ij} + b_{ij}$.
• Multiplication by a Scalar: Multiplying matrix $A$ by scalar $k$ multiplies every element of $A$ by $k$. $kA = [ka_{ij}]$.
• Negative of a Matrix: Denoted by $-A = (-1)A$.
• Difference: $A - B = A + (-1)B$.
• Multiplication of Matrices: The product $AB$ is defined ONLY if the number of columns of $A$ equals the number of rows of $B$.
  • If $A$ is $m \times n$ and $B$ is $n \times p$, then $C = AB$ is of order $m \times p$.
  • JEE TIPMatrix multiplication is a row-by-column operation. The $(i, k)^{th}$ element of $C$ is the sum of the products of corresponding elements of the $i^{th}$ row of $A$ and the $k^{th}$ column of $B$.

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

• Symmetric Matrix: A square matrix where $A' = A$ (i.e., $a_{ij} = a_{ji}$).
• Skew-Symmetric Matrix: A square matrix where $A' = -A$ (i.e., $a_{ij} = -a_{ji}$).JEE TIPPutting $i=j$ yields $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies a_{ii} = 0$. Thus, all diagonal elements of a skew-symmetric matrix are strictly zero.
• Orthogonal Matrix:JEE TIPA square matrix $A$ is orthogonal if $A A' = A' A = I$.

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 $A$ ($A^{-1}$).

• Rectangular matrices cannot possess an inverse because products $AB$ and $BA$ wouldn't be of the same order.
• Inverse of a square matrix, if it exists, is unique.

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

Conditions & Limitations

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 ($|A| \neq 0$).

Standard Derivations & Step-by-Step Problem Solving

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 $n \in \mathbb{N}$.

Previous Year JEE Topics

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 TIPIf $A = I + B$ 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 TIPThe 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 $A^{-1}$.

Memory Aids & JEE Traps

• Trap 1 - The Commutativity Algebraic Fallacy:

  • Misconception: Expanding a matrix binomial expression like $(A+B)(A-B)$ instantly yields the standard scalar identity $A^2 - B^2$.
  • Correct Understanding: Matrix multiplication is strictly non-commutative ($\mathbf{AB \neq BA}$) in general. Distributing the binomial yields $(A+B)(A-B) = A^2 - AB + BA - B^2$. Unless the problem explicitly states that matrices $A$ and $B$ commute, the middle cross-product terms cannot be canceled out.

• Trap 2 - Linear Product Transpose Order:

  • Misconception: Evaluating the transpose of a multi-matrix product expands linearly without changing positions, such as $(ABC)^T = A^T B^T C^T$.
  • Correct Understanding: The reversal law of transposes mandates that the product order must be completely reversed upon distribution: $(ABC)^T = C^T B^T A^T$. Applying this property without inverting the sequence creates completely non-conformable dimensions or invalid matrix products.

• Trap 3 - Skew-Symmetric Diagonal Non-Zero Elements:

  • Misconception: A matrix is classified as skew-symmetric as long as its off-diagonal elements satisfy $a_{ij} = -a_{ji}$, leaving the diagonal elements free to take any value.
  • Correct Understanding: By definition, a skew-symmetric matrix satisfies $A^T = -A$, which forces the elements along the main diagonal to satisfy $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies \mathbf{a_{ii} = 0}$. The principal diagonal of a real skew-symmetric matrix must consist entirely of zeros.

• Trap 4 - The Identity Matrix Power Expansion:

  • Misconception: Computing high-degree powers of the identity matrix scales its inner elements up or expands the matrix size linearly ($I^n = nI$).
  • Correct Understanding: The identity matrix acts as the multiplicative identity element, meaning $I^n = I$ remains true for any integer power $n$, and $AI = IA = A$. While you can treat it like the scalar number $1$ inside matrix polynomial expansions, you must still strictly preserve matrix dimension conformability and multiplication sequence constraints.

• Trap 5 - Zero Matrix Product Cancellation:

  • Misconception: If the square of a matrix or the product of two matrices results in a null matrix ($X^2 = O$), then the matrix $X$ itself must be a null matrix ($X = O$).
  • Correct Understanding: Matrix algebra allows for non-zero divisors of zero, known as nilpotent matrices. For example, the non-zero matrix $X = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ yields a perfect null matrix product $X^2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O$ even though $X \neq O$.

• Trap 6 - Multiplicative Trace Splitting:

  • Misconception: The matrix trace operator splits across multiplication identically to addition, satisfying $\text{Tr}(AB) = \text{Tr}(A) \times \text{Tr}(B)$.
  • Correct Understanding: The trace operator is linear over matrix addition ($\text{Tr}(A+B) = \text{Tr}(A) + \text{Tr}(B)$) but completely non-linear over multiplication. However, it satisfies the critical cyclic permutation invariance property: $\text{Tr}(ABC) = \text{Tr}(BCA) = \text{Tr}(CAB)$. Note that general non-cyclic permutations are not equal ($\text{Tr}(ABC) \neq \text{Tr}(BAC)$).

• Trap 7 - Determinant Scalar Pull-Out Scaling:

  • Misconception: Pulling a scalar constant multiplier $k$ out of a determinant operation changes its value linearly, satisfying $|kA| = k|A|$.
  • Correct Understanding: A determinant factor scales row by row. If a square matrix $A$ has dimensions of order $n \times n$, pulling the scalar factor out of the entire matrix pulls it out of all $n$ rows simultaneously, yielding the relation: $|kA| = k^n|A|$. Forgetting this power of $n$ factor is a major cause of numerical errors in JEE matrix problems.

• Trap 8 - Symmetry Profile of Product Transposes:

  • Misconception: The product of a matrix and its transpose ($AA^T$) can yield a skew-symmetric matrix depending on the entries of $A$.
  • Correct Understanding: The product forms $AA^T$ and $A^T A$ are universally and unconditionally symmetric for any real matrix $A$, regardless of whether $A$ itself is square, rectangular, symmetric, or asymmetric. This is verified by checking their transposes using the reversal law: $(AA^T)^T = (A^T)^T A^T = \mathbf{AA^T}$.

• Trap 9 - Non-Invertible Matrix Cancellation:

  • Misconception: If two matrix products share a common leading matrix factor, you can cancel it directly from both sides: $AB = AC \implies B = C$.
  • Correct Understanding: This cancellation operation is strictly valid if and only if matrix $A$ is non-singular ($|A| \neq 0$), meaning its inverse $A^{-1}$ exists. If $A$ is a singular or non-square matrix, the product equation can balance perfectly even when $B \neq C$. You cannot divide or cancel matrices blindly.

• Trap 10 - Entries Permutation Matrix Multiplicity:

  • Misconception: Finding the total number of unique $3 \times 3$ matrices that can be formed using entries chosen strictly from the set $\{0, 1\}$ is evaluated by multiplying choices by positions ($2 \times 9 = 18$).
  • Correct Understanding: A $3 \times 3$ matrix contains exactly 9 distinct entry positions. Because each individual position can be filled independently using any of the 2 available choices, evaluating the total configuration space requires applying the fundamental multiplication principle of combinations: $\text{Total Matrices} = 2 \times 2 \times \dots \times 2 = 2^9 = \mathbf{512}$.