Matrices & transformations lab

See what a matrix actually does. Drag the two columns of a 2 × 2 matrix and watch the whole plane stretch, turn and flip, chain two transformations together, then multiply matrices entry by entry, expand determinants, find inverses and solve systems of equations by row reduction, the matrix method and Cramer's rule.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 12 — What a matrix is and how to add and multiply

Concept: a matrix is a rectangle of numbers; an m × n matrix has m rows and n columns. You add matrices of the same order entry by entry, and multiply by a number entry by entry. To multiply AB, the entry in row i, column j is (row i of A) · (column j of B), so A needs as many columns as B has rows.

Open Arithmetic and press a 2 × 2 example with A × B. Blue marks a row of A and orange a column of B.
Use Show the working for the entry in to move through the four entries and read each row · column sum.
Press 2 × 3 times 3 × 2: the answer is 2 × 2, because inner numbers match and outer numbers give the order.
Change B to a 2 × 3 matrix and see Not defined appear: the inner numbers no longer match.
Try A + B with different orders to see why addition needs the same order.

Takeaway: AB exists when (columns of A) = (rows of B), and its order is (rows of A) × (columns of B).

Class 12 — Multiplication is not commutative

Concept: for matrices, AB and BA are usually different, and can even have different orders. A product can also be the zero matrix without either factor being zero. But matrix multiplication is still associative, and it distributes over addition.

Press AB ≠ BA, then switch A × B to B × A and compare the two answers.
Press a product of two non-zero matrices that is zero: both factors are non-zero, yet AB = 0.
Press multiplying by the identity: AI = A.
Open Composition and swap the order of rotate and shear: the picture changes, which is the same fact seen as geometry.
Press transpose of a 2 × 3 and check that rows become columns.

Takeaway: AB ≠ BA in general, AB = 0 does not force A = 0 or B = 0, and (AB)ᵀ = BᵀAᵀ.

Class 12 — Determinants

Concept: the determinant is a number attached to a square matrix. For a 2 × 2 matrix it is ad − bc. For 3 × 3 you expand along any row or column, adding each entry times its cofactor (the minor with the sign + − + −).

Open Determinant & inverse with the 3 × 3 preset and read the expansion along row 1 in the working.
Change Expand the determinant along to another row or a column: the terms change but the total does not.
Press Swap R1 and R2 and watch the determinant change sign; press Multiply R1 by 2 and watch it double.
Press R1 → R1 + 2·R2: the determinant does not change.
Press a singular 3 × 3: the determinant is 0, so the rows are dependent.

Takeaway: swapping two rows flips the sign, scaling a row scales the determinant, adding a multiple of a row changes nothing, and det = 0 means no inverse.

Class 12 — Adjoint and inverse

Concept: the adjoint of A is the transpose of its cofactor matrix. If det A ≠ 0 then A⁻¹ = adj A ÷ det A, and A · A⁻¹ = I. A matrix with det A = 0 is singular and has no inverse.

In Determinant & inverse press 2 × 2 and read the cofactors, the adjoint and the inverse.
Check the card A · adj A = (det A) I shows a tick.
Press a singular 2 × 2: there is no inverse.
Choose the 3 × 3 preset and read every cofactor, with its sign.
Try triangular: its determinant is the product of the diagonal entries.

Takeaway: A⁻¹ = adj A ÷ det A, and it exists only when det A ≠ 0.

Class 12 — Solving systems of linear equations

Concept: write the equations as AX = B. If det A ≠ 0 there is exactly one solution, X = A⁻¹B (or Cramer's rule). If det A = 0, use row reduction: the system is inconsistent (no solution) when the ranks of A and [A | b] differ, and has infinitely many solutions when they are equal but less than the number of unknowns.

Open Linear systems with x + y = 3, x − y = 1 and read x = 2, y = 1 from all three methods.
Slide Row-reduction step to watch the augmented matrix turn into the identity one operation at a time.
Press parallel lines: no solution and see the two lines on the graph never meet.
Press the same line: infinitely many and read the general solution with a free variable.
Try the three 3 × 3 presets: unique, none and infinitely many.

Takeaway: compare the rank of A with the rank of [A | b] to know which of the three cases you are in.

Class 12 — Matrices as transformations (beyond the core syllabus)

Concept: a 2 × 2 matrix sends every point (x, y) to a new point. Its first column is where (1, 0) lands and its second column is where (0, 1) lands. The determinant is how much areas are scaled, and its sign tells you whether the plane is flipped. This is not in the NCERT chapter but it shows why the matrix rules exist.

Open Transformations and drag the teal arrow tip, the image of î, then the orange one, the image of ĵ.
Press rotate 90°, scale × 2, shear and reflect in y = x and read the matrix and the determinant each time.
Drag the green point P and check P′ = MP against the working.
Tick the eigenvector lines and see the directions that are only stretched, never turned.
Press squash onto the x-axis: the determinant is 0, the plane collapses to a line and there is no inverse.

Takeaway: the columns of a matrix show where the basis vectors go, and the determinant is the area scale factor.

The maths behind it

What each mode covers

Transformations — drag the columns of a 2 × 2 matrix and watch a shape, the grid and the unit square change. Determinant as area scale factor, inverse, eigenvector directions and P′ = MP. An extension beyond the class 12 syllabus that explains why the rules work.

Composition — do one transformation and then another (rotate, stretch, shear, reflect, project) and see the single matrix BA that does both, and how the answer changes with the order.

Arithmetic — add, subtract, scale, multiply and transpose matrices up to 3 × 3, with exact fractions and every entry's row · column working. Class 12 matrices.

Determinant & inverse — expand along any row or column, cofactors, adjoint, inverse, and the effect of row operations on the determinant. Class 12 determinants.

Linear systems — the same system solved by row reduction (step by step), the matrix method X = A⁻¹B and Cramer's rule, with the consistency test using rank. Class 12 matrices and determinants.

Part of Shunya. See the syllabus map for the chapters these modes map to. All matrix arithmetic here is exact (fractions), so answers such as 1/3 are not rounded. The plane transformations use decimals.

Formulas used in this lab

Matrices: order and types Class 12

Order of a matrix: $m \times n$
Identity matrix: $AI = IA = A$
Symmetric: $A^T = A$
Skew-symmetric: $A^T = -A$

Matrix arithmetic Class 12

Sum (same order): $(A + B)_{ij} = a_{ij} + b_{ij}$
Scalar multiple: $(kA)_{ij} = k\,a_{ij}$
Product entry: $(AB)_{ij} = \sum_k a_{ik}b_{kj}$
Order of a product: $(m \times n)(n \times p) = m \times p$
Not commutative: $AB \ne BA \text{ in general}$

Transpose Class 12

Definition: $(A^T)_{ij} = a_{ji}$
Rules: $(A^T)^T = A,\qquad (A + B)^T = A^T + B^T$
Transpose of a product: $(AB)^T = B^TA^T$
Transpose of a scalar multiple: $(kA)^T = kA^T$

Determinant Class 12

2 × 2: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
Expansion along row i: $\det A = \sum_j a_{ij}C_{ij}$
Cofactor: $C_{ij} = (-1)^{i+j} M_{ij}$
Product rule: $\det(AB) = \det A \cdot \det B$
Scalar multiple (n × n): $\det(kA) = k^n \det A$
Transpose: $\det(A^T) = \det A$

Row operations and the determinant Class 12

Swap two rows: $\det \to -\det$
Multiply a row by k: $\det \to k\det$
Add a multiple of a row to another: $\det \text{ unchanged}$
Two equal rows, or a row of zeros: $\det = 0$

Adjoint and inverse Class 12

Adjoint: $\operatorname{adj} A = (C_{ij})^T$
Key identity: $A\,(\operatorname{adj} A) = (\det A)\,I$
Inverse: $A^{-1} = \dfrac{\operatorname{adj} A}{\det A},\qquad \det A \ne 0$
2 × 2 inverse: $\dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
Inverse of a product: $(AB)^{-1} = B^{-1}A^{-1}$

Solving linear systems Class 12

Matrix form: $AX = B$
Matrix method (det A ≠ 0): $X = A^{-1}B$
Cramer's rule: $x = \dfrac{D_x}{D},\quad y = \dfrac{D_y}{D},\quad z = \dfrac{D_z}{D}$
One solution: $\text{rank } A = \text{rank } [A|B] = n$
No solution: $\text{rank } A \lt \text{rank } [A|B]$
Infinitely many: $\text{rank } A = \text{rank } [A|B] \lt n$

Matrices as transformations (beyond the core syllabus) Class 12

Image of a point: $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$
Rotation by θ: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Area scale factor: $|\det M|$
Eigenvalues: $\lambda^2 - (\operatorname{tr} M)\lambda + \det M = 0$
Composition (B after A): $M = BA$