Matrices
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Order, Types and Equality
Part of: Anatomy of a Matrix
A matrix is a rectangle of numbers with m rows and n columns, holding mn entries. One row or one column makes row and column matrices; equal rows and columns make square matrices, whose special citizens are diagonal, scalar, identity and zero matrices. Two matrices match only with the same order and the same entry everywhere.
Why it matters. Order decides every later operation, and equality turns matrix equations into small puzzles.
Formulas
- An m by n matrix holds mn elements
- Square needs m = n
- Equal matrices match in order and every entry
Key words
- Order of a matrix
- : The size tag m by n, rows first then columns.
- Square matrix
- : A matrix with equal rows and columns.
- Diagonal matrix
- : A square matrix with zeros everywhere off the diagonal.
- Scalar matrix
- : A diagonal matrix with one repeated diagonal value.
- Identity matrix
- : A scalar matrix with ones down the diagonal.
- Zero matrix
- : A matrix with every entry zero.
Try it in two minutes
- Count rows and columns of three printed matrices.
- Sort them into square and non-square piles.
- Match equal entries to solve for two unknowns.
Try at home. List all possible orders of a 12-element matrix.
Addition and Scalar Multiplication
Part of: Anatomy of a Matrix
Matrices add entry by entry, but only when orders match exactly. A scalar stretches every entry equally. Addition commutes and groups freely, scalars distribute over sums, and paired sum-difference equations split apart by halving.
Why it matters. Combined tables and scaled bills in shops and surveys are exactly these operations.
Formulas
- Add only matching orders, entry by entry
- k(A + B) is kA + kB
- Solve X + Y and X − Y systems by halving sum and difference
Key words
- Matrix addition
- : Adding matching entries of two same-order matrices.
- Scalar multiplication
- : Multiplying every entry of a matrix by one number.
- Distributive law
- : The rule that k(A + B) is kA + kB.
- Compatible order
- : Matching sizes that allow an operation.
Try it in two minutes
- Add two small same-order grids entry by entry.
- Double one grid and check each entry doubled.
- Solve an X + Y and X − Y pair by halving.
Try at home. Build September and October sale grids and find the combined grid.
Matrix Multiplication
Part of: Multiplying and Flipping Matrices
A product AB exists only when columns of A match rows of B, and each entry pairs a row of A with a column of B. The result is m × p for m × n times n × p. Order matters: AB and BA can differ or one can fail entirely, and nonzero matrices can multiply to zero.
Why it matters. Bills from rate tables and multi-step transformations are matrix products in disguise.
Formulas
- AB needs columns of A to match rows of B
- AB of m by n with n by p is m by p
- Square pairs can still refuse to commute
Key words
- Matrix product
- : The row-by-column weave of two compatible matrices.
- Order rule
- : Columns of the first must equal rows of the second.
- Zero divisor
- : A nonzero matrix multiplying another nonzero matrix to zero.
- Multiplicative identity
- : The identity grid leaving square matrices unchanged.
Try it in two minutes
- Check the middle numbers match before multiplying.
- Pair the first row of A with each column of B.
- Swap the order and compare the two results.
Try at home. Price two shopping lists with a rate column and compare totals.
Transpose, Symmetry and Inverses
Part of: Multiplying and Flipping Matrices
Flipping rows into columns gives the transpose, which reverses product order. Symmetric grids mirror themselves while skew grids mirror with a sign flip and carry zeros down the diagonal. Every square grid splits into symmetric and skew halves, and a partner undoing both products is the unique inverse.
Why it matters. Symmetry halves the work in physics and statistics, and inverses solve linear systems.
Formulas
- Transpose of a product reverses the factor order
- A plus its transpose is symmetric; A minus is skew
- Inverse needs AB and BA both identity
Key words
- Transpose
- : The grid got by swapping rows with columns.
- Symmetric matrix
- : A square grid unchanged by transpose.
- Skew-symmetric matrix
- : A square grid sign-flipped by transpose, with zero diagonal.
- Inverse matrix
- : The unique partner giving identity products both ways.
Try it in two minutes
- Flip a small grid across its diagonal on paper.
- Check whether the flip matches or sign-flips.
- Multiply a claimed inverse pair both ways.
Try at home. Split [[2, 5], [1, 4]] into symmetric and skew halves.
What your child will practise
- Addition and Scalar Multiplication: Desert Quick Fire (Quick Fire)
- Matrix Multiplication: Jungle Puzzle (Clue Puzzle)
- Matrix Multiplication: Space Quest (Flip and Match)
- Order, Types and Equality: Ocean Quick Fire (Quick Fire)
- Transpose, Symmetry and Inverses: Desert Puzzle (Clue Puzzle)
- Quiz: Addition and Scalar Multiplication (8 questions)
- Quiz: Matrix Multiplication (8 questions)
- Quiz: Order, Types and Equality (8 questions)
- Quiz: Transpose, Symmetry and Inverses (8 questions)