Matrices — Class 12 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Matrices, written for Class 12 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
Matrices in one look
row and column moves that build an inverse two ways, then solve equations with it
Elementary Transformations
- six elementary transformations: three on rows, three on columns, each giving a matrix equivalent, never equal
- interchange rows i and j: R_i ↔ R_j, also written R_ij
- the same move on columns: C_i ↔ C_j, written C_ij
- scale row i by k≠0: R_i to kR_i; same on columns, C_i to kC_i
- add k times row i to row j: R_j to R_j+kR_i, only R_j changes
- the symbol ~ means equivalent to, never equal: write A ~ B, not A = B
Notation for Elementary Transformations
- interchange rows i and j
R_i ↔ R_j - scale row i by k, k≠0
R_i to kR_i - add k times row i to row j
R_j to R_j+kR_i
Inverse by Transformation
- if AB=BA=I for square A, B is the inverse A^-1; when it exists, it is unique
- for AA^-1=I: augment with I, reduce left to I; same moves turn right into A^-1
- for A^-1A=I instead, use column transformations throughout; never mix rows and columns in one reduction
- a row (or column) collapsing entirely to zero before I is reached means A is singular
Inverse by Transformation, worked
- augment A with I: left side has rows (2,1) and (1,1), right side has rows (1,0) and (0,1)
- R_1 ↔ R_2: left rows become (1,1) and (2,1), right rows become (0,1) and (1,0)
- R_2 to R_2-2R_1: left rows become (1,1) and (0,-1), right rows become (0,1) and (1,-2)
- R_2 to -R_2: left rows become (1,1) and (0,1), right rows become (0,1) and (-1,2)
- R_1 to R_1-R_2: left side is now I, right rows are (1,-1) and (-1,2)
- **A^-1 has rows (1,-1) and (-1,2)**
Minors and Cofactors
- minor M_ij of a_ij: the determinant left after deleting row i and column j
- cofactor A_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)
Page 1 of 5. The full sheet, all 5 pages, opens free in the app.
What this Matrices sheet covers
- Elementary Transformations
- Notation for Elementary Transformations
- interchange rows i and j
- scale row i by k, k≠0
- add k times row i to row j
- Elementary Transformation Method
- Inverse by Transformation
- Inverse by Transformation, worked
- Minors and Cofactors
- Adjoint method
- Inverse by Adjoint
- adjoint, the transpose of the cofactor matrix
- inverse from the adjoint, needs |A|≠0
- inverse of a product, order reverses
- product of A with its own adjoint
- Special Cases: Singular, Identity, Null, Diagonal
- Adjoint by Cofactors, worked
- Inverse by Adjoint, worked
- determinant of A, rows (2,-3) and (4,1)
- Elementary Transformation Method vs Adjoint Method
- By Transformation
- By Adjoint
- Quick Reference: Every Key Formula
- adjoint
- inverse from the adjoint
- A times its own adjoint
- inverse of a product
- determinant of the adjoint
- adjoint of the adjoint
- solving AX=B
- Solving Equations by Matrices
- Method of Inversion
- Method of Inversion, worked
- Symbols
- Method of Reduction
- Method of Reduction, worked
- Method of Inversion vs Method of Reduction
- Inversion
- Reduction
- 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 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.