Determinants
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Evaluating Determinants
Part of: Values and Geometry of Determinants
A determinant is a single number owned only by square matrices. Two-by-two uses cross products, three-by-three expands along any row or column with alternating signs, and zero-rich lines expand fastest. Scaling a matrix scales its determinant by powers of the order.
Why it matters. Determinants test invertibility, measure areas and solve linear systems.
Formulas
- 2 by 2 determinant is ad minus bc
- Expand along the line with most zeros
- |kA| is k to the order times |A|
Key words
- Determinant
- : The single number associated with a square matrix.
- Expansion
- : Unfolding a determinant along one row or column.
- Order of a determinant
- : The size n of its n by n matrix.
- Scaling rule
- : Multiplying a matrix by k multiplies its determinant by k to the order.
Try it in two minutes
- Compute a 2 × 2 determinant by cross products.
- Expand a 3 × 3 along its emptiest line.
- Solve one determinant equation for x.
Try at home. Verify that doubling a 3 × 3 matrix multiplies its determinant by 8.
Area, Minors and Cofactors
Part of: Values and Geometry of Determinants
Three vertices feed a 3 × 3 determinant whose half absolute value is the triangle area; zero area means collinear points. A minor deletes one row and column, a cofactor adds the checkerboard sign, and a determinant equals any row dotted with its own cofactors, while alien rows sum to zero.
Why it matters. Area tests and cofactor expansions run through board papers every year.
Formulas
- Area is half the absolute determinant of vertices
- Zero area means collinear points
- A row dotted with alien cofactors sums to zero
Key words
- Minor
- : The smaller determinant left after deleting one row and column.
- Cofactor
- : A minor dressed with its plus or minus checkerboard sign.
- Collinear points
- : Points on one line, giving zero triangle area.
- Absolute area rule
- : Area always takes the positive half of the determinant.
Try it in two minutes
- Plot three points and feed them into the area determinant.
- Cover one row and column to read a minor.
- Dot a row with its cofactors to rebuild the value.
Try at home. Test (0, 0), (4, 0) and (0, 3) for area 6 with the determinant.
Adjoint and Inverse of a Matrix
Part of: Inverses and Linear Systems
The adjoint is the transposed grid of cofactors, and a matrix times its adjoint gives the determinant times identity. Zero determinant means singular with no inverse; nonzero means invertible with inverse equal to adjoint over determinant. The adjoint of a product reverses order, just like transpose.
Why it matters. The adjoint formula is the gateway to solving linear systems by matrices.
Formulas
- A times adj A is det times I
- Invertible matches non-singular exactly
- |adj A| is |A| to the power n minus 1
Key words
- Adjoint
- : The transpose of the cofactor grid of a matrix.
- Singular matrix
- : A square matrix with zero determinant and no inverse.
- Non-singular matrix
- : A square matrix with nonzero determinant, hence invertible.
- Inverse formula
- : The rule that A inverse is adjoint over determinant.
Try it in two minutes
- Find all four cofactors of a 2 × 2 grid.
- Transpose the cofactor grid into the adjoint.
- Divide by the determinant to land the inverse.
Try at home. Find the inverse of [[4, 1], [3, 2]] with the adjoint shortcut.
Solving Linear Systems
Part of: Inverses and Linear Systems
Equations pack into AX = B, and a non-singular coefficient grid gives the unique answer X = A inverse times B. A zero determinant hands the verdict to (adj A)B: nonzero means no solution, while zero leaves infinitely many or none to separate.
Why it matters. The matrix method solves coupled bills, mixtures and circuits in one stroke.
Formulas
- Non-singular A gives X as A inverse times B
- Zero det with nonzero (adj A)B means no solution
- Zero det with zero (adj A)B needs deeper check
Key words
- Matrix method
- : Solving AX = B through X = A inverse times B.
- Consistent system
- : Equations owning at least one solution.
- Inconsistent system
- : Equations owning no solution at all.
- Unique solution
- : Exactly one answer, from a non-singular grid.
Try it in two minutes
- Pack two equations into A, X and B grids.
- Test the determinant for zero or nonzero.
- Premultiply by the inverse where it exists.
Try at home. Solve x + y = 5 and 2x − y = 4 by the matrix method.
What your child will practise
- Adjoint and Inverse of a Matrix: Desert Quick Fire (Quick Fire)
- Adjoint and Inverse of a Matrix: Ocean Puzzle (Clue Puzzle)
- Area, Minors and Cofactors: Jungle Puzzle (Clue Puzzle)
- Evaluating Determinants: Ocean Quest (Flip and Match)
- Solving Linear Systems: Jungle Quick Fire (Quick Fire)
- Quiz: Adjoint and Inverse of a Matrix (8 questions)
- Quiz: Area, Minors and Cofactors (8 questions)
- Quiz: Evaluating Determinants (8 questions)
- Quiz: Solving Linear Systems (8 questions)