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CBSE Class 12 Maths

Determinants

Each concept is a short lesson, then questions until you get three right.

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10 concepts, a short lesson each

  • Determinants0/3
  • Adjoint0/4
  • Linear equations0/3

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Value of a determinant

∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc; a 3×33 \times 3 expands along a row with signs +−++ - +

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Determinants0 of 3

  1. Value of a determinant∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc; a 3×33 \times 3 expands along a row with signs +−++ - +Start
  2. Minors and cofactorsCofactor Aij=(−1)i+jMijA_{ij} = (-1)^{i+j}M_{ij}: the minor with the checkerboard signStart
  3. Area of a triangle and collinearityArea =12∣∣x1y11x2y21x3y31∣∣= \frac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|; 00 means the points are collinearStart

Adjoint0 of 4

  1. Adjoint of a matrixadj⁡A\operatorname{adj}A is the transpose of the cofactor matrix; for 2×22 \times 2, swap the diagonal and negate the other twoStart
  2. Singular and invertible matricesAA has an inverse exactly when ∣A∣≠0|A| \ne 0; a matrix with ∣A∣=0|A| = 0 is singularStart
  3. Inverse by the adjoint methodA−1=1∣A∣adj⁡AA^{-1} = \dfrac{1}{|A|}\operatorname{adj}A; check with AA−1=IAA^{-1} = IStart
  4. Properties of the inverse and adjointA(adj⁡A)=∣A∣IA(\operatorname{adj}A) = |A|I; (AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}; ∣adj⁡A∣=∣A∣n−1|\operatorname{adj}A| = |A|^{n-1}; (A−1)−1=A(A^{-1})^{-1} = AStart

Linear equations0 of 3

  1. Matrix equationsAX=BAX = B gives X=A−1BX = A^{-1}B, not BA−1BA^{-1}: the inverse goes on the side where AA wasStart
  2. Solving equations by inversionWrite the system as AX=BAX = B, find A−1A^{-1}, then X=A−1BX = A^{-1}B gives every unknown at onceStart
  3. Word problems as linear systemsName the unknowns, write one equation per condition, solve by inversion and answer in the problem's unitsStart

Practise Determinants

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions