Matrices are a major Class 12 topic and the foundation of engineering mathematics. Enter a 2×2 or 3×3 matrix row by row to get its determinant, inverse and transpose with the working you would write in an exam.
Choose 2×2 or 3×3.
Enter the entries row by row, separated by commas.
Read the determinant, inverse and transpose.
2×2 determinant: |A| = ad − bc
Inverse: A⁻¹ = adj(A) ÷ |A|
For a square matrix of size 2 × 2 or 3 × 3, this calculator returns three things: the determinant, the inverse and the transpose. For a 3 × 3 matrix it shows the determinant by expansion along the first row and builds the inverse by the adjoint method, exactly as done in the NCERT Class 12 matrices and determinants chapters.
Matrices store sets of numbers in rows and columns, and they underpin systems of equations, computer graphics transformations, economics input-output models, engineering structures and machine learning. The determinant tells you whether a matrix can be inverted, and the inverse lets you undo a transformation or solve AX = B. The transpose, which swaps rows and columns, is needed for the adjoint and for many formulas in statistics. Entries are typed row by row, separated by commas.
1. For a 2 × 2 matrix [a b; c d], the determinant is ad − bc.
2. If the determinant is not zero, the 2 × 2 inverse is (1 ÷ (ad − bc)) × [d −b; −c a]: swap a and d, negate b and c, then divide.
3. For a 3 × 3 matrix, expand along row 1: det = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃, where each M is the 2 × 2 minor left after deleting that row and column.
4. Find all nine cofactors Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ.
5. Transpose the cofactor matrix to get the adjoint, adj(A).
6. Divide the adjoint by the determinant to get A⁻¹.
7. Find the transpose separately by writing each row of A as a column.
8. Check that A × A⁻¹ gives the identity matrix.
Geometrically, a 2 × 2 matrix transforms the unit square into a parallelogram, and the determinant is that parallelogram's signed area. For 3 × 3, it is the signed volume of the transformed unit cube. A determinant of 0 means the shape has been squashed flat into a line or plane, so information is lost and cannot be recovered. That is exactly why a matrix with zero determinant, called singular, has no inverse. A negative determinant means orientation has been flipped, like a mirror image.
Multiplying A by its adjoint gives a matrix whose diagonal entries are expansions of det(A) along each row, and whose off-diagonal entries are expansions using the wrong row's cofactors. Those off-diagonal sums equal the determinant of a matrix with two identical rows, which is zero. So A × adj(A) = det(A) × I, and dividing by det(A) gives A⁻¹ = adj(A) ÷ det(A). The 2 × 2 swap-and-negate rule is this same formula written out.
The transpose does not change the determinant: det(Aᵀ) = det(A). The determinant of a product is the product of determinants, det(AB) = det(A) det(B), and det(A⁻¹) = 1 ÷ det(A). The inverse of a product reverses order: (AB)⁻¹ = B⁻¹A⁻¹. Swapping two rows changes the determinant's sign, and multiplying a row by k multiplies it by k. The calculator uses floating-point arithmetic, so a determinant that should be exactly zero can occasionally appear as a tiny non-zero number.
Riya is checking her Class 12 homework, where she found the inverse of the 2 × 2 matrix with rows 4, 7 and 2, 6 by hand.
Determinant: ad − bc = 4×6 − 7×2 = 10
Inverse = (1/det) × [d, −b; −c, a]: [ 0.6, -0.7 ] [ -0.2, 0.4 ]
Answer: Determinant 10; Inverse [ 0.6, -0.7 ] [ -0.2, 0.4 ]; Transpose [ 4, 2 ] [ 7, 6 ]
Forgetting the alternating signs of cofactors, especially the minus on the middle term of row 1.
Using the cofactor matrix itself as the adjoint without transposing it.
For 2 × 2, negating a and d instead of swapping them, and swapping b and c instead of negating them.
Entering the matrix column by column instead of row by row.
Trying to invert a matrix whose determinant is zero, or dividing by the determinant twice.
Checking Class 12 answers for determinants, adjoints and inverses.
Solving systems of linear equations by the matrix method X = A⁻¹B.
Undoing 2D and 3D transformations in computer graphics.
Finding the area of a triangle from coordinates using a determinant.
Testing whether a set of vectors is linearly independent.
Why does my matrix have no inverse?
Its determinant is zero, which means its rows are linearly dependent.
How can I check the inverse?
Multiply A by its inverse; the result should be the identity matrix.