Determinants and Matrices — Class 11 Maharashtra HSC Mathematics revision notes
A 6-page visual revision sheet for Determinants and Matrices, written for Class 11 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Determinants and Matrices in one look
a single number squeezed out of a square array, and the array itself as an object you can add, scale and multiply
A Determinant, Order 2 and Order 3
- a determinant of order 2 is the square array a,b,c,d between vertical bars
- its value is ad-bc
- a determinant of order 3 has 9 elements in 3 rows and 3 columns
- denoted a_ij for the entry in row i, column j
- a determinant is written with a Capital letter or the symbol Δ
- its array sits between vertical BARS, not square brackets
- brackets are the notation for a matrix instead
- there are six ways to expand a 3x3 determinant, one per row and per column
- expanding along any of them gives the same value
Expansion Along the First Row
- order 2
|a b; c d|=ad-bc - order 3, by row 1
D=a_11C_11+a_12C_12+a_13C_13
Minors and Cofactors
- the minor M_ij of element a_ij is the determinant left after deleting row i, column j
- the cofactor C_ij=(-1)^i+jM_ij: same value as the minor, sign flipped when i+j is odd
- signs alternate in a checkerboard starting with plus at position (1,1)
- for a 2x2 determinant a,b,c,d: the minor of a is d, of b is c
- the minor of c is b, the minor of d is a
Q. Find the minors and cofactors of every element of |1 2; 3 4|
- minor of a_11=1 is the surviving entry a_22=4, so M_11=4 and C_11=(-1)^1+1(4)=4
- minor of a_12=2 is a_21=3, so M_12=3 and C_12=(-1)^1+2(3)=-3
- minor of a_21=3 is a_12=2, so M_21=2 and C_21=(-1)^2+1(2)=-2
- minor of a_22=4 is a_11=1, so M_22=1 and C_22=(-1)^2+2(1)=1
Answer: C_11=4, C_12=-3, C_21=-2, C_22=1
Page 1 of 6. The full sheet, all 6 pages, opens free in the app.
What this Determinants and Matrices sheet covers
- A Determinant, Order 2 and Order 3
- Expansion Along the First Row
- order 2
- order 3, by row 1
- Minors and Cofactors
- Properties of Determinants
- Main Diagonal and Triangular Determinants
- Cramer's Rule, Three Equations Three Unknowns
- eq. 1
- eq. 2
- eq. 3
- solution, D≠0
- Cramer's Rule, by D
- Consistency of Three Equations in Two Variables
- Area of a Triangle and Collinearity
- area of triangle ABC
- A Matrix, and its Order
- Types of Matrices
- Row / Column
- Zero (Null)
- Square
- Diagonal
- Scalar
- Identity In
- Upper / Lower Triangular
- Symmetric
- Skew-Symmetric
- Singular / Non-singular
- Transpose
- definition
- Algebra of Matrices
- Multiplication of Two Matrices
- conformable when
- A_m× n· B_n× p=C_m× p
- Properties of the Transpose
- Symmetric vs Skew-Symmetric
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Determinants and Matrices the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.