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Determinants and Matrices

Ch. 4Std 11

Easy Overview

Imagine you have a bunch of numbers and you want to pack them neatly into a box so you can do maths on them all at once. That's a matrix. A determinant is just a special number that tells you something about that box — like whether it's squishy or not. Sounds abstract, but this stuff powers computer graphics, physics, and even your phone's camera filters.

What even is a matrix?

A matrix is just a rectangular grid of numbers. Think of it like a spreadsheet — rows and columns. A 2×3 matrix has 2 rows and 3 columns. You can add matrices if they're the same size (just add matching entries). Multiply by a scalar (a number) and every entry gets multiplied. That's the simple stuff.

Matrix multiplication — the weird one

Multiplying two matrices isn't just 'multiply matching numbers'. You take the rows of the first and columns of the second. Element (i,j) = row i of first × column j of second — dot product style. If A is m×n and B is n×p, the result is m×p. The inner dimensions must match, or it's a no-go. And AB ≠ BA most of the time. Yeah, matrix multiplication doesn't commute.

Determinants — what are they?

A determinant is a single number calculated from a square matrix. For a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc. If it's 0, the matrix is 'singular' — think of it like a flat rubber band that's lost its spring. Non-zero determinant means the matrix is invertible. For 3×3, you expand using minors and cofactors. It's a bit tedious but the pattern is mechanical.

Properties of determinants

There are rules that make life easier. If two rows are swapped, the determinant flips sign. If two rows are identical, determinant = 0. If you multiply a row by a constant k, the determinant gets multiplied by k. Adding a multiple of one row to another doesn't change the determinant. These properties help you compute determinants faster without expanding fully.

Inverse of a matrix

The inverse of a matrix A is A⁻¹, such that A × A⁻¹ = I (the identity matrix). For 2×2, A⁻¹ = (1/det(A)) × adj(A). The adjoint is just switching a and d, negating b and c — easy. If det(A) = 0, no inverse exists. The inverse is useful for solving systems of equations: Ax = b → x = A⁻¹b.

Key Points

  • •Matrix: rectangular array of numbers with m rows, n columns
  • •Addition: same dimensions, add matching entries
  • •Multiplication: AB exists if columns(A) = rows(B)
  • •Determinant of 2×2: ad − bc
  • •det(AB) = det(A)·det(B)
  • •A is invertible iff det(A) ≠ 0
  • •Inverse of 2×2: (1/det) × adj — swap a,d and negate b,c
  • •Use Cramer's rule or matrix inverse to solve linear systems

Practice Questions

  • Find the determinant of a 3×3 matrix using expansion of minors.
  • If A = [[1, 2], [3, 4]], find A⁻¹ and verify A·A⁻¹ = I.
  • Prove that the determinant of a matrix with two equal rows is 0.
  • Solve the system: 2x + 3y = 8, x − y = −1 using matrix method.
  • If A and B are 2×2 matrices, show that det(AB) = det(A)·det(B) with an example.