Matrix Calculator - Punjabi Numerals

Input: A = [[2,1],[5,3]], B = [[1,2],[3,4]]

Result: A × B = [[2×1+1×3, 2×2+1×4],[5×1+3×3, 5×2+3×4]] = [[5,8],[14,22]]

What can the matrix calculator do?

The matrix calculator supports addition, subtraction, multiplication, transpose, determinant, inverse, RREF (reduced row echelon form), eigenvalues, trace, and solving linear systems (Ax = B). You can work with matrices up to 10×10, including rectangular matrices. The calculator shows step-by-step solutions for RREF, determinant, and inverse operations.

Why is my matrix inverse showing an error?

A matrix has no inverse when its determinant is zero — this is called a singular matrix. This happens when one row (or column) is a linear combination of other rows, meaning the matrix does not have full rank. The calculator will display an error in this case because the inverse does not exist. Check your matrix for duplicate or proportional rows, which are the most common cause.

What does it mean if the determinant of a 3x3 matrix is zero?

A zero determinant means the matrix is singular — it has no inverse and the system of equations it represents has either no solution or infinitely many solutions. In practical terms, if you are solving Ax = b and det(A) = 0, the system is either inconsistent (no solution) or dependent (infinitely many solutions). A non-zero determinant means the system has exactly one unique solution. The calculator computes the determinant of any square matrix including 3x3 matrices.

How do I use the matrix multiplication calculator?

Enter your first matrix (e.g., a 2×3 matrix) as matrix A, then enter your second matrix as matrix B. For multiplication to work, the number of columns in the first matrix must equal the number of rows in the second matrix. Select "Multiply" and the calculator computes the result matrix. The calculator shows the step-by-step row-by-column dot products when steps are enabled.

How do I find the inverse of a matrix online?

Enter a square matrix (same number of rows and columns) as matrix A and select "Inverse". The calculator computes the inverse using Gauss-Jordan elimination if it exists. If the determinant is zero, the matrix is singular and has no inverse — the calculator will notify you. The steps show the augmented matrix [A | I] being reduced to [I | A⁻¹].

How do I transpose a matrix?

Transposing flips a matrix over its diagonal, converting rows to columns and vice versa. Enter your matrix and select "Transpose". For example, a 2×3 matrix becomes a 3×2 matrix where the element at position (i,j) moves to position (j,i).

What size matrices can I work with?

The calculator supports matrices of various sizes, typically up to 10×10. You can create rectangular matrices (different row and column counts) for operations like addition and multiplication, or square matrices for operations like determinant and inverse.

Can I add matrices of different sizes?

No, matrix addition and subtraction require matrices of the same dimensions (same number of rows and columns). If you try to add mismatched matrices, the calculator will display an error. Make sure both matrices have identical dimensions before adding or subtracting.

Does the matrix calculator show steps?

Yes. Enable the "Show Steps" toggle before performing an operation. The calculator displays step-by-step solutions for RREF (showing each row operation), determinant (showing the cofactor expansion), and inverse (showing the Gauss-Jordan elimination process with the augmented matrix [A | I]).

How do I solve a 3x3 matrix?

Enter a 3×3 matrix and choose the operation you need. For determinant, the calculator uses cofactor expansion along the first row. For inverse, it forms the augmented matrix [A | I] and performs row operations until A becomes the identity. For RREF, it performs Gaussian elimination to reduce the matrix to its simplest form. For eigenvalues, it solves the characteristic polynomial det(A - λI) = 0.

How do I use the matrix solver for a system of equations?

Enter the coefficient matrix as matrix A and the constant vector as matrix B, then select "Solve Linear System". The calculator solves Ax = B using the inverse method (x = A⁻¹B) or Gaussian elimination. For example, to solve 2x + y = 7 and x - y = 2, enter A = [[2,1],[1,-1]] and B = [[7],[2]], and the solver returns x = 3, y = 1.

Can I compute RREF (reduced row echelon form) on this calculator?

Yes. Enter a matrix and select "RREF". The calculator performs Gaussian elimination to reduce the matrix so that each leading entry is 1, each leading 1 is the only non-zero entry in its column, and each leading 1 appears to the right of the one above it. RREF is useful for solving systems of equations, finding the rank of a matrix, and determining linear independence. Steps are available showing each row operation.

Can I compute eigenvalues on this matrix calculator?

Yes. Enter a square matrix and select "Eigenvalues". The calculator solves the characteristic equation det(A - λI) = 0 to find the eigenvalues. Eigenvalues are scalar values λ where Av = λv for some eigenvector v. The calculator displays both real and complex eigenvalues. Note that eigenvalue computation is available for matrix A.

Is this the same numeral system used in Pakistani Punjab?

No. Pakistani Punjab primarily uses Shahmukhi script (a Perso-Arabic script), which uses Eastern Arabic/Persian numerals. The Gurmukhi numerals on this calculator are for Indian Punjabi written in the Gurmukhi script.