MATRICES JEE Notes PDF | Free MCQs & Study Material Download
MATRICES – COMPLETE STUDY NOTES WITH SOLVED PROBLEMS AND PRACTICE QUESTIONS Free Download
Introduction
Matrices are one of the most important topics in mathematics, forming the foundation for linear algebra and its applications. A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. The word matrix comes from the Latin word meaning womb, symbolizing that many mathematical ideas originate here.
Matrices are widely used in engineering, physics, computer graphics, economics, and statistics. In school and college mathematics, matrices are mainly used to
Solve simultaneous linear equations
Represent linear transformations
Work with determinants and inverses
Definition
A matrix is written as
A = [a11 a12 ... a1n; a21 a22 ... a2n; ... ; am1 am2 ... amn]
Here
aij = element in the i-th row and j-th column
m = number of rows
n = number of columns
Order of the matrix = m × n
Types of Matrices
1. Row Matrix – Only one row. Example [1 2 3]
2. Column Matrix – Only one column. Example [1;2;3]
3. Square Matrix – Same number of rows and columns
4. Diagonal Matrix – Non-zero elements only on the main diagonal
5. Scalar Matrix – Diagonal elements all equal
6. Identity Matrix (I) – Diagonal elements = 1, rest = 0
7. Zero or Null Matrix – All elements = 0
8. Symmetric Matrix – A = A^T
9. Skew-Symmetric Matrix – A^T = -A
Basic Operations
1. Addition and Subtraction
Possible only if matrices have the same order. Add or subtract elements position-wise.
A + B = [aij + bij]
2. Scalar Multiplication
Each element is multiplied by the same constant k.
kA = [k × aij]
3. Matrix Multiplication
If A(m × n) and B(n × p), then AB = C(m × p)
Where cij = Σ aik bkj over k=1 to n
Note Multiplication is not commutative i.e. AB ≠ BA in general
4. Transpose of a Matrix
The transpose of A is A^T where rows become columns and columns become rows.
Example A = [1 2;3 4], A^T = [1 3;2 4]
5. Determinant
For a 2×2 matrix
|A| = |a b;c d| = ad − bc
For 3×3,
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
6. Adjoint and Inverse
Adjoint (Adj A) Transpose of cofactor matrix
Inverse (A⁻¹) A⁻¹ = Adj(A)/|A| if |A| ≠ 0
Property A × A⁻¹ = A⁻¹ × A = I
Solved Examples
Example 1 Addition
Find A + B if A = [1 2;3 4], B = [5 6;7 8]
Solution
A + B = [1+5 2+6;3+7 4+8] = [6 8;10 12]
Example 2 Transpose
Find A^T for A = [1 2 3;4 5 6]
Solution
A^T = [1 4;2 5;3 6]
Example 3 Determinant
Find |A| for A = [2 3;5 7]
Solution
|A| = (2)(7) − (3)(5) = 14 − 15 = −1
Example 4 Matrix Multiplication
If A = [1 2;3 4], B = [2 0;1 2] find AB.
Solution
AB = [1×2+2×1 1×0+2×2;3×2+4×1 3×0+4×2] = [4 4;10 8]
Practice Questions
1. If A = [2 3;4 1] and B = [1 2;3 5], find A − B.
2. Find the transpose of [1 2 3;4 5 6].
3. Multiply [1 2;3 4] and [2 1;0 3].
4. Find the determinant of [3 2;5 4].
5. Find the inverse of [1 2;3 4].
6. Verify whether A^T = A for A = [2 3;3 2].
7. Show that AB ≠ BA using suitable matrices.
8. If A = [1 1;1 1], find A².
9. For A = [a 0;0 b], find |A|.
10. Find the product AB where A = [1 0 2;−1 3 1], B = [3 1;2 1;1 0].