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CBSE Class 12 · Mathematics

Determinants

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Determinants” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Determinant

A determinant is a number from a square matrix as taught. For [[a,b],[c,d]] it is ad−bc. |2 1; 1 3| = 6−1 = 5. Det is a number, not a second matrix. A non-square has no det here.

Adding the entries as the det is a miss.

Figure. The 2-by-2 determinant is the main-diagonal product minus the other-diagonal product: ad − bc.

How it works

  1. Need squareThe gate.
  2. Form ad−bc (or the taught 3×3 expand)The number.
  3. Keep it a number, not an arrayDet ≠ matrix.

2×2 det

Find |2 1; 1 3|.

  • ad2×3=6
  • bc1×1=1
  • ad−bc5

Pro tip. Two-by-two: ad minus bc.

2Area of a Triangle

Area of a triangle with vertices (x1,y1), … is (1/2)|det of the taught 3×3| as framed. Area is a positive number; the det may need an absolute value. (0,0),(4,0),(0,6) gives 12.

Dropping the 1/2 is a common miss.

Figure. The determinant formula for three vertices is the signed area of this triangle. Base times height over two is the same number when the base sits on an axis.

How it works

  1. Write the taught 3×3 from the verticesThe array.
  2. Take |det|/2 as taughtThe area.
  3. Keep it positiveA size.

3Minors and Cofactors

Minors and cofactors: Mᵢⱼ is the det after deleting row i, column j; Cᵢⱼ = (−1)^{i+j} Mᵢⱼ as taught. Cofactor is a signed minor. One 2×2 minor from a 3×3 is enough.

Calling the leftover entries “the minor” without a det is a miss.

Figure. The minor of a11 is the 2-by-2 left after that row and column are struck. The cofactor attaches the sign (−1)^{i+j}.

How it works

  1. Delete the row and columnThe leftover.
  2. Take that det, then the sign (−1)^{i+j}Minor, then cofactor.
  3. Keep i,j as 1-based as the lesson wrote themHonest.

4Adjoint and Inverse of a Matrix

Adjoint is the transpose of the cofactor matrix; A⁻¹ = (1/det A) adj A when det ≠ 0 as taught. Adjoint is an array; inverse uses it. det 0 means no inverse here.

Writing adj A as Aᵀ is a skip of cofactors.

Figure. The inverse exists only when det A is not zero. Then A inverse is (1/det A) times the adjoint.

How it works

  1. Build cofactors, then transposeadj A.
  2. Divide by det if det ≠ 0The inverse.
  3. Refuse inverse when det is 0The gate.

5Applications of Determinants and Matrices

Applications: solve AX=B by A⁻¹B or Cramer as taught. Application is a system-solve, not a second det-definition. One 2×2 system is enough.

A pretty inverse with no system is not this heading.

Figure. Two independent linear equations are two lines. One crossing is one solution — the pair Cramer’s rule names.

How it works

  1. Write the system as AX=BThe given.
  2. Use A⁻¹B or Cramer as taughtThe application.
  3. Check by plugging backHonest.

6A definition is a test you can run

Determinant-word is a test: ad−bc, an area, a minor, or adj/inverse. If you only say “bars”, you have a heading.

A matrix-box is not the test.

Figure. The definition is a computation you can run: 1×4 − 2×3 = −2. Non-zero means an inverse exists.

How it works

  1. Name the number or the adj writeThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

7Name the given before the unknown

The given is a square array. The unknown is the number or the inverse. Copy [[2,1],[1,3]] before you compute 2+3 as det.

Using a+d as det is a silent swap.

Figure. Write every entry you were given. The determinant is the number those entries produce — it is not a second unknown hiding in the grid.

How it works

  1. Copy the squareThe given.
  2. Name det, area, or inverseThe unknown.
  3. Then computeGiven first.

8One worked case is enough at this class

One worked case is enough: det=5, then if asked inverse uses 1/5. Do not stack four 3×3 expands unless the lesson did. The case teaches ad−bc.

A poster of ten dets is not more science.

Figure. One 2-by-2 expansion is enough to hold the pattern: main product, other product, subtract.

How it works

  1. Take one 2×2The case.
  2. Form ad−bcThe teach.
  3. Stop — one caseThis class.
|2 1; 1 3| is
  1. 5
  2. 6
  3. 3

ad−bc.

Notes

  • The official chapter title is “Determinants”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • |a b; c d| = ad−bc
  • A⁻¹=(1/det A) adj A if det≠0

Recap

Hold these pegs from the official chapter “Determinants”. The wording is ExamMaster’s teaching, not a textbook recap.

Determinant
A determinant is a number from a square matrix as taught.
Area of a Triangle
Area of a triangle with vertices (x1,y1), … is (1/2)|det of the taught 3×3| as framed.
Minors and Cofactors
Minors and cofactors: Mᵢⱼ is the det after deleting row i, column j; Cᵢⱼ = (−1)^{i+j} Mᵢⱼ as taught.
Adjoint and Inverse of a Matrix
Adjoint is the transpose of the cofactor matrix; A⁻¹ = (1/det A) adj A when det ≠ 0 as taught.
Applications of Determinants and Matrices
Applications: solve AX=B by A⁻¹B or Cramer as taught.
A definition is a test you can run
Determinant-word is a test: ad−bc, an area, a minor, or adj/inverse.

Practise Determinants

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