Determinants ā Class 12 CBSE Mathematics revision notes
A 3-page visual revision sheet for Determinants, written for Class 12 CBSE 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 in one look
one number per square array, and it decides whether the array can be undone
A determinant, defined
- order 2, entries a, b, c, d read across each row
|A| = ad-bc - order 3, general entry in row i column j
a_ij, 1ā¤ i,jā¤3 - only a square array has one at all
|A|_nĆ n
Properties that shortcut the arithmetic
- Swap two rows: the determinant flips sign
- Two rows identical: the determinant is exactly 0
- Common factor in a row: pull it out in front of the determinant
- Triangular array: the determinant is just the product of the diagonal entries
More properties worth knowing
- swapping two columns behaves exactly like swapping two rows: the sign flips
- multiply one row by k: the whole determinant scales by k, once, not per entry
- adding a multiple of one column to another leaves the determinant unchanged too, like rows
- the determinant of A transpose equals the determinant of A itself: |A^T| = |A|
Expand a 3Ć3 determinant
Q. Evaluate the determinant of the array with rows (2, 3, 1), (1, -1, 2) and (0, 4, -3).
- expand along row 1: |A| = 2M_11 - 3M_12 + 1M_13, sign pattern (+, -, +)
- M_11 = (-1)(-3) - (2)(4) = 3-8=-5
- M_12 = (1)(-3) - (2)(0) = -3-0=-3
- M_13 = (1)(4) - (-1)(0) = 4-0=4
- |A| = 2(-5) - 3(-3) + 1(4) = -10+9+4
Answer: |A| = 3
Page 1 of 3. The full sheet, all 3 pages, opens free in the app.
What this Determinants sheet covers
- A determinant, defined
- order 2, entries a, b, c, d read across each row
- order 3, general entry in row i column j
- only a square array has one at all
- Properties that shortcut the arithmetic
- Swap two rows
- Two rows identical
- Common factor in a row
- Triangular array
- More properties worth knowing
- Expand a 3Ć3 determinant
- Minors and cofactors
- minor of entry a_ij: delete its row and column, take the determinant of what remains
- cofactor of entry a_ij
- expansion along any row i (every row gives the same value)
- Expanding along a different row or column
- Adjoint and inverse
- adjoint of A: transpose the array of cofactors
- A times its own adjoint, either order
- inverse, when it exists
- When does A inverse actually exist
- Singular vs non-singular
- Singular, |A| = 0
- Non-singular, |A| not 0
- Area of a triangle from its vertices
- vertices (x1,y1), (x2,y2), (x3,y3); D is that array's determinant
- the three points are collinear exactly when
- Test three points for collinearity
- Cramer's rule for three variables
- system a1x+b1y+c1z=d1 and two more rows shaped like it
- D uses the coefficients; D1, D2, D3 each replace one column with the constants d1, d2, d3
- Solving AX=B: what |A| tells you
- Putting the chapter back together
- 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 the sheet opens inside the study block itself ā 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.