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Determinants: CBSE Class 12 Maths knowledge organiser

Everything to know about determinants on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Determinant
A number found from a square matrix; for a 2 × 2 matrix it is ad − bc.
Adjoint
The transpose of the matrix of cofactors.
Inverse
A⁻¹ = adj A / |A|, which exists only when |A| ≠ 0.

Key formulas

\(2\times2\)\(\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc\)
Minor; cofactor\(M_{ij};\) \(A_{ij}=(-1)^{i+j}M_{ij}\)
\(|A|=\sum_j a_{ij}A_{ij}\) along any row (or column); elements × cofactors of a different row sum to 0
Area of triangle\(\tfrac12\left|\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}\right|;\) \(\text{collinear}\iff\text{det}=0\)
Adjoint = transpose of the cofactor matrix\(A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I\)
Inverse\(A^{-1}=\frac{1}{|A|}\operatorname{adj}A\) \((|A|\ne0)\)
Order \(n\)\(|AB|=|A||B|,\) \(|A^T|=|A|,\) \(|kA|=k^n|A|,\) \(|\operatorname{adj}A|=|A|^{n-1},\) \(|A^{-1}|=\tfrac1{|A|}\)
\(AX=B\)\(|A|\ne0:\ X=A^{-1}B\ \text{(unique)}\)
\(|A|=0\): \((\operatorname{adj}A)B\ne O\) ⇒ inconsistent; \((\operatorname{adj}A)B=O\) ⇒ infinitely many or none

Worked example

Area of the triangle with vertices \((0, 0)\), \((4, 1)\), \((1, 3)\): \(\dfrac12\begin{vmatrix}0&0&1\\4&1&1\\1&3&1\end{vmatrix} = \dfrac12(12 - 1) = \dfrac{11}{2}\) sq units.

Common mistakes

  • Sign pattern of cofactors \(\begin{smallmatrix}+&-&+\\-&+&-\\+&-&+\end{smallmatrix}\) forgotten, especially for \(A_{12}, A_{21}, A_{23}, A_{32}\).
  • Writing the matrix of cofactors instead of its transpose as \(\mathrm{adj}\,A\).
  • \(|kA| = k|A|\) instead of \(k^n|A|\).
  • Forgetting the modulus (or one of the two cases) in area problems.

You should be able to…

  • Expand 2x2 and 3x3 determinants, find minors and cofactors, and find inverses of 2x2 matrices.
  • Use determinants for area, collinearity and lines; use |kA|, |adj A| and inverse results; decide consistency of small systems.
  • Score full marks on the 5-mark matrix-method question and on determinant case studies, with every step mark visible.
  • Handle parameter systems (unique / none / infinitely many), adj-inverse identities and unfamiliar applications.

The printable sheet

Determinants knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Determinants knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers