Matrices and Determinants | IISER Smart Prep

1. Concept of Matrices

A matrix is an ordered rectangular array of numbers or functions, called elements or entries. A matrix having m rows and n columns is called a matrix of order m × n.

Notation: A = [a_ij]_m×n, where i = 1, 2,..., m and j = 1, 2,..., n.
Equality: Two matrices are equal if they have the same order and their corresponding elements are equal.

Types of Matrices:

Column/Row Matrix: Matrix with one column (m × 1) or one row (1 × n).
Square Matrix: Number of rows equals columns (m = n).
Diagonal Matrix: Square matrix where all non-diagonal elements are zero.
Scalar & Identity (I): Diagonal matrix with identical diagonal elements. If diagonal elements are 1, it's an Identity matrix.
Zero Matrix (O): All elements are zero.

2. Algebra of Matrices

Addition & Scalar Multiplication

Addition (A + B): Possible only if A and B have the same order. Add corresponding elements. It is commutative (A+B=B+A) and associative.
Scalar Multiplication (kA): Every element of A is multiplied by the scalar k.
Negative of a Matrix: -A = (-1)A.

AB ≠ BA (Generally)

Matrix multiplication is not commutative.

Multiplication of Matrices (AB)

To compute A × B, the columns of A must equal the rows of B. If A is (m × n) and B is (n × p), matrix AB will be of order (m × p).

Associative Law: A(BC) = (AB)C.
Distributive Law: A(B + C) = AB + AC.
Identity Property: AI = IA = A.

3. Transpose & Matrix Ops

The Transpose (A^T or A') is obtained by interchanging rows into columns. For A = [a_ij]_m×n, A^T = [a_ji]_n×m.

(A^T)^T = A
(A + B)^T = A^T + B^T
(AB)^T = B^TA^T (Reversal law is crucial!)

Symmetric and Skew-Symmetric

Symmetric: A^T = A. (Symmetrical across the main diagonal).
Skew-symmetric: A^T = -A. (Diagonal elements are zero).
Theorem: Any square matrix can be written as the sum of a symmetric and a skew-symmetric matrix: A = ½(A + A^T) + ½(A - A^T).

Elementary Operations:

Transforming rows (or columns) algebraically:

Interchanging any two rows/columns (R_i ↔ R_j).
Multiplying a row/column by a non-zero scalar (R_i → kR_i).
Adding a scalar multiple of another row/column (R_i → R_i + kR_j).

4. Determinants

Every square matrix A = [a_ij] has a number called its Determinant, denoted as |A| or det(A).

1x1 Matrix: det(A) = a₁₁
2x2 Matrix: |A| = (a₁₁)(a₂₂) - (a₂₁)(a₁₂)
3x3 Matrix: Expanded using row or column cofactors.

Expansion of Determinant

Along row i:

|A| = Σ a_ij A_ij

Along column j:

|A| = Σ a_ij A_ij

Where A_ij is the cofactor of element a_ij.

Determinant Properties:

|AB| = |A| |B|.
|A^T| = |A|.
If any two rows/columns are identical, |A| = 0.
Interchanging two rows/columns multiplies the determinant by -1.
|kA| = kⁿ|A|, where n is the order of the square matrix.

Area of a Triangle:

Using vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃):

Area = ½ | [x₁ y₁ 1]
[x₂ y₂ 1]
[x₃ y₃ 1] |

If Area = 0, the three points are collinear.

5. Adjoint and Inverse

Minors and Cofactors

Minor (M_ij): Determinant obtained by deleting the i^th row and j^th column.
Cofactor (A_ij): A_ij = (-1)^i+j M_ij.

The Adjoint (adj A) is the transpose of the cofactor matrix.

A(adj A) = (adj A)A = |A| I
|adj A| = |A|^n-1 (Crucial for IAT!)
adj(AB) = (adj B)(adj A)

Invertible Matrix

A square matrix A is invertible if there exists a matrix A^-1 such that:

AA^-1 = A^-1A = I

This happens if and only if |A| ≠ 0.

Uniqueness of Inverse

If A^-1 exists, then it is unique. There cannot be two different inverses for the same matrix.

A^-1 = (adj A) / |A|

Inverse exists if and only if |A| ≠ 0 (A is non-singular).

6. Systems of Linear Equations

Matrices can be used to solve systems of linear equations: AX = B.

Consistency of System

|A| ≠ 0 → Unique solution (X = A^-1B).
|A| = 0 and rank(A) = rank([A|B]) → Infinite solutions.
|A| = 0 and rank(A) ≠ rank([A|B]) → No solution.

7. IAT Exam Focus Points

High-Yield Tips:

When solving 3x3 matrices, use elementary operations to create zeros before expanding the determinant.
Orthogonal Matrices: A A^T = I. For these matrices, A^-1 = A^T.
Idempotent Matrices: A² = A. Trace is equal to the rank.
The determinant represents the area scale factor of the transformation.

8. Practice Mock Test

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