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MH HSC 12 Maths

Matrices

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

0/11

11 concepts, a short lesson each

  • Transformations0/2
  • Adjoint0/5
  • Linear equations0/4

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Elementary transformations

Interchange two rows, multiply a row by k≠0k \ne 0, add kk times one row to another; the same three for columns

Start

Transformations0 of 2

  1. Elementary transformationsInterchange two rows, multiply a row by k≠0k \ne 0, add kk times one row to another; the same three for columnsStart
  2. Inverse by elementary transformationsWrite A=IAA = IA and apply row operations until I=BAI = BA (or A=AIA = AI with column operations); never mix rows and columnsStart

Adjoint0 of 5

  1. Minors and cofactorsMijM_{ij} deletes row ii and column jj; Aij=(−1)i+jMijA_{ij} = (-1)^{i+j}M_{ij}; expanding along a row gives ∣A∣|A|Start
  2. 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
  3. Singular and invertible matricesAA has an inverse exactly when ∣A∣≠0|A| \ne 0; a matrix with ∣A∣=0|A| = 0 is singularStart
  4. Inverse by the adjoint methodA−1=1∣A∣adj⁡AA^{-1} = \dfrac{1}{|A|}\operatorname{adj}A; check with AA−1=IAA^{-1} = IStart
  5. 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 4

  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. Solving equations by reductionRow-reduce the system to a triangular form, then solve from the last equation upwardsStart
  4. Word problems as linear systemsName the unknowns, write one equation per condition, solve by inversion and answer in the problem's unitsStart

Practise Matrices

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

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