How this instrument works
A² means A multiplied by itself using ordinary matrix multiplication — rows of A run against columns of A, exactly the way any two matrices combine. That is a genuinely different operation from squaring each entry on its own, and different again from multiplying the whole matrix by the plain number 2, which is what a separate matrix-by-scalar calculator computes. The shared appearance of a '2' is coincidental: one operation runs A through itself as a transformation, the other just rescales every entry by a constant.
A useful shortcut, from the Cayley–Hamilton theorem, is that every 2×2 matrix satisfies its own characteristic equation: A² = trace(A)·A − det(A)·I, where I is the identity matrix. That means A² can be built from just two scalars — the same trace and determinant a characteristic-polynomial calculation already reports — without ever multiplying a row against a column. It is a small, surprising fact: the whole squared result is hiding inside two plain numbers most people compute for an entirely different reason.
A genuine edge case shows how far matrix squaring departs from ordinary arithmetic: with real numbers, only 0² equals 0, but one can be nonzero and still square to all zeros. Take [[0, 1], [0, 0]]: it is not all zeros, yet multiplying it by itself returns [[0, 0], [0, 0]] exactly. Such a matrix is called nilpotent, and it represents a transformation so degenerate that applying it twice erases whatever the first application left behind.
- Enter the top row of your matrix into row1,col1 and row1,col2 — these are a and b in [[a, b], [c, d]].
- Enter the bottom row into row2,col1 and row2,col2 — c and d complete it.
- Read A²: row1,col1 and A²: row1,col2 for the top row of A².
- Read A²: row2,col1 and A²: row2,col2 for its bottom row.
- Change just one entry, such as b or c, and rerun — because multiplication mixes rows against columns, every one of the four A² outputs shifts, not only the entry you touched.
Worked example — squaring [[1, 2], [3, 4]]
Set row1,col1 = 1, row1,col2 = 2, row2,col1 = 3, and row2,col2 = 4, the matrix [[1, 2], [3, 4]]. Row against column: A²: row1,col1 = a² + bc = 1×1 + 2×3 = 1 + 6 = 7, and A²: row1,col2 = ab + bd = 1×2 + 2×4 = 2 + 8 = 10. The bottom row follows the same rule: A²: row2,col1 = ca + dc = 3×1 + 4×3 = 3 + 12 = 15, and A²: row2,col2 = cb + d² = 3×2 + 4×4 = 6 + 16 = 22 — so A² = [[7, 10], [15, 22]]. Squaring each entry instead would have wrongly produced [[1, 4], [9, 16]], an unrelated result that shares not one digit with this one.
The Cayley–Hamilton shortcut checks the same answer without a single row-times-column multiplication: trace(A) = 1 + 4 = 5 and det(A) = 1×4 − 2×3 = −2, so A² = trace(A)·A − det(A)·I = 5×[[1, 2], [3, 4]] − (−2)×I = [[5, 10], [15, 20]] + [[2, 0], [0, 2]] = [[7, 10], [15, 22]]. Both routes land on the identical answer, because A² represents the same linear transformation applied twice, however it gets computed.
Questions
Does squaring a matrix mean squaring each of its entries?
No. A² means A multiplied by itself with full row-times-column multiplication — rows against columns — not (a², b², c², d²). For A = [[1, 2], [3, 4]], A² is [[7, 10], [15, 22]]; squaring each entry separately would wrongly give [[1, 4], [9, 16]], the result of an entirely different, invalid operation.
Is 'matrix squared' the same thing as 'matrix times two'?
No, though the shared number 2 causes real confusion. A² multiplies A by itself using row-times-column multiplication; 2A instead multiplies every entry by the scalar 2, which is what a separate matrix-by-scalar calculator computes. For [[1, 2], [3, 4]], A² is [[7, 10], [15, 22]] while 2A is only [[2, 4], [6, 8]] — unrelated answers from unrelated operations.
Is there a faster way to find A² than multiplying rows by columns?
Yes, for any 2×2 matrix. The Cayley–Hamilton theorem guarantees A² = trace(A)·A − det(A)·I, so two scalars — the same trace and determinant a characteristic-polynomial calculation reports — rebuild the whole squared matrix. For [[1, 2], [3, 4]], trace = 5 and det = −2, giving A² = 5A + 2I = [[7, 10], [15, 22]], matching the direct multiplication exactly.
Can a nonzero matrix still square to all zeros?
Yes — something impossible for ordinary numbers, where only 0² equals 0. The nonzero matrix [[0, 1], [0, 0]] squares to [[0, 0], [0, 0]] exactly. Such a matrix is called nilpotent: applying its transformation twice collapses every vector to the origin, even though a single application does not.
What does squaring a matrix mean geometrically?
It means applying A's transformation twice in a row on the same input. A rotation matrix squared rotates by double the angle; a scaling map squared scales each direction by the square of its original factor. A² is not a separate kind of operation — it is A run again on whatever A already produced.
How do the eigenvalues of A² relate to the eigenvalues of A?
Each eigenvalue of A² is the square of the matching eigenvalue of A. The 90° rotation matrix [[0, −1], [1, 0]] has eigenvalues ±i; squaring it gives [[−1, 0], [0, −1]], whose eigenvalue is −1 twice — exactly i² and (−i)², and exactly what a 180° rotation should produce.