Matrix Calculator
Add, subtract and multiply 2x2 matrices, plus determinant values.
Overview
The Matrix Calculator handles the basic operations on small matrices: addition, subtraction, multiplication and determinant. Enter the entries of two 2x2 matrices (or use it with one) and the calculator returns the sum, difference, product and the determinant of each.
It is built for students learning linear algebra, programmers debugging 2D transforms and engineers running quick consistency checks on a worked problem. The 2x2 case is small enough to do by hand but easy to slip up on — this tool keeps you honest.
How it works
Matrix addition and subtraction are element-wise: (A ± B)_ij = A_ij ± B_ij. Multiplication of A * B uses the row-by-column rule: each entry of the product is the dot product of a row of A with a column of B.
For a 2x2 matrix [[a, b], [c, d]] the determinant is a*d - b*c. The determinant of a product equals the product of determinants: det(A * B) = det(A) * det(B). Both halves come out as part of the same calculation.
Examples
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]
→ A + B = [[6, 8], [10, 12]]
A * B → [[19, 22], [43, 50]]
det(A) = 1*4 - 2*3 = -2
det(B) = 5*8 - 6*7 = -2, det(A * B) = (-2) * (-2) = 4
FAQ
Does matrix multiplication commute?
No. Generally A * B != B * A. Order matters.
What does the determinant tell me?
For 2x2 it's the signed area scaling factor of the transformation. Zero means the transformation collapses the plane to a line.
Can I do 3x3 or larger?
This tool focuses on 2x2. For 3x3 system solving, see Linear System 3x3.
Why is the product of determinants equal to the determinant of the product?
It's a fundamental theorem of linear algebra — det is a homomorphism from invertible matrices to non-zero reals.
Can I add or subtract matrices of different sizes?
No — dimensions must match. Multiplication requires the column count of the left matrix to equal the row count of the right matrix.