A matrix is a rectangular arrangement of numbers (called elements or entries) organised in rows and columns, enclosed in square brackets. Matrices are fundamental tools in mathematics, used to represent systems of linear equations, transformations, and much more.
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Key Concepts
Order of a Matrix: An m x n matrix has m rows and n columns. The element in the i-th row and j-th column is denoted aij.
- Types of Matrices
- Row matrix: Only 1 row (1 x n).
- Column matrix: Only 1 column (m x 1).
- Square matrix: Same number of rows and columns (n x n).
- Zero (null) matrix: All elements are zero, denoted O.
- Identity matrix: Square matrix with 1s on the main diagonal and 0s elsewhere, denoted In.
- Diagonal matrix: A square matrix where all off-diagonal elements are zero.
- Symmetric matrix: AT = A (aij = aji for all i, j).
- Skew-symmetric matrix: AT = -A (aij = -aji; diagonal elements are all 0).
Transpose of a Matrix: The matrix obtained by interchanging rows and columns of A, denoted AT.
- Matrix Operations
- Addition/Subtraction: Matrices of the same order; add/subtract corresponding elements.
- Scalar Multiplication: Multiply every element by the scalar.
- Matrix Multiplication: A (m x n) and B (n x p) can be multiplied to give AB of order m x p. (AB)ij = sum of aik · bkj for k = 1 to n.
- Key Properties of Multiplication:
- AB is not necessarily equal to BA (not commutative in general).
- A(BC) = (AB)C (associative).
- A(B + C) = AB + AC (distributive).
- AI = IA = A (identity).
Result: Every square matrix A can be expressed as sum of a symmetric and a skew-symmetric matrix: A = (A + AT)/2 + (A - AT)/2.
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Worked Examples
If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find A + B.
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].
Find AB where A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]].
AB = [[1 · 2 + 2 · 1, 1 · 0 + 2 · 3], [3 · 2 + 4 · 1, 3 · 0 + 4 · 3]] = [[4, 6], [10, 12]].
Find the transpose of A = [[1, 2, 3], [4, 5, 6]].
AT = [[1, 4], [2, 5], [3, 6]] (rows become columns).
Show that A = [[0, 3, -1], [-3, 0, 2], [1, -2, 0]] is skew-symmetric.
AT = [[0, -3, 1], [3, 0, -2], [-1, 2, 0]] = -A. So AT = -A, confirming skew-symmetry.
If 2A + B = [[3, 1], [2, 4]] and A = [[1, 0], [1, 2]], find B.
2A = [[2, 0], [2, 4]]. B = [[3,1],[2,4]] - [[2,0],[2,4]] = [[1,1],[0,0]].
Express A = [[2, 3], [1, -1]] as the sum of symmetric and skew-symmetric matrices.
P = (A + AT)/2 = ([[2,3],[1,-1]] + [[2,1],[3,-1]])/2 = [[2, 2], [2, -1]].
Q = (A - AT)/2 = ([[2,3],[1,-1]] - [[2,1],[3,-1]])/2 = [[0, 1], [-1, 0]].
Check: P + Q = [[2,3],[1,-1]] = A.
If A = [[2, 1], [1, 3]] and I is the 2x2 identity, find A2 - 5A + 4I.
A2 = [[5, 5], [5, 10]]. 5A = [[10, 5], [5, 15]]. 4I = [[4,0],[0,4]].
A2 - 5A + 4I = [[5-10+4, 5-5+0], [5-5+0, 10-15+4]] = [[-1, 0], [0, -1]] = -I.
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Common mistakes
- Trying to multiply A (m x n) by B (p x q) when n is not equal to p — multiplication is only defined when inner dimensions match.
- Assuming AB = BA; this is generally false for matrices.
- Forgetting that the element aij in aij refers to row i, column j (not the other way around).
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Summary
Matrices provide a compact notation for linear data. Matrix addition is straightforward, but multiplication is non-commutative. Transpose, symmetric, and skew-symmetric matrices are key concepts tested in board exams.