Perform matrix addition, subtraction, multiplication, transpose, or find the determinant, for 2x2 and 3x3 matrices.
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Matrix A
Matrix B
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Addition and subtraction combine two matrices of the same size by adding or subtracting corresponding entries. Multiplication combines two square matrices of the same size using the standard row-by-column dot product rule. The determinant is a single number computed from a square matrix that indicates, among other things, whether the matrix is invertible (a zero determinant means it isn't). Transpose flips a matrix over its diagonal, turning rows into columns.
These are by far the most commonly needed sizes for coursework, introductory linear algebra, and quick manual checks. Larger matrices are typically handled with dedicated computational software rather than manual entry, since the number of cells to enter grows quickly with size.
For Matrix A = [[1, 2], [3, 4]] and Matrix B = [[5, 6], [7, 8]]: addition gives [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]. Multiplication (row-by-column) gives [[1×5+2×7, 1×6+2×8], [3×5+4×7, 3×6+4×8]] = [[19, 22], [43, 50]]. The determinant of A alone is (1×4) − (2×3) = 4 − 6 = −2, and its transpose swaps rows and columns to give [[1, 3], [2, 4]].
Multiplying matrices entry-by-entry like addition, matrix multiplication uses the row-by-column dot product rule instead, so A×B ≠ [[1×5, 2×6], [3×7, 4×8]], that pattern is a common but incorrect shortcut.
Assuming matrix multiplication is commutative (A×B = B×A), it generally isn't, order matters and multiplying in the reverse order typically produces a different result.
Forgetting that only square matrices (equal rows and columns) have a determinant, and that this calculator's determinant and transpose operations act on Matrix A only, changes to Matrix B won't affect those results.
This calculator requires both matrices to be the same size (2x2 or 3x3) for all operations. General matrix multiplication does allow different sizes as long as the number of columns in the first matches the number of rows in the second, but that flexibility isn't supported here.
A zero determinant means the matrix is 'singular', it doesn't have an inverse, and represents a transformation that collapses space into a lower dimension (for example, flattening a 2D plane into a line).
No, only addition, subtraction and multiplication use Matrix B. Determinant and transpose only operate on Matrix A, so Matrix B's fields are hidden for those operations.
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