Exact fractions or decimals
With exact fractions on, every entry is kept as a fraction, so the inverse of a matrix with whole numbers comes out as clean values like −1/5 instead of −0.19999. Turn it off to work in decimals. If a cell holds something irrational, such as sqrt(2) or pi, the calculator switches to decimals for that calculation and says so.
Determinant, inverse and RREF
The determinant is found by row reducing to an upper triangular matrix and multiplying the diagonal, keeping track of row swaps, which flip the sign. The inverse comes from reducing [A | I] until the left side is the identity. RREF, reduced row echelon form, shows every swap, scaling and elimination, which is also how you read off the rank and solve systems of equations.
Eigenvalues and LU
Eigenvalues are the roots of the characteristic polynomial det(λI − A). In exact mode you get the polynomial and exact roots whenever they exist, like (5 ± √33)/2. Otherwise they are computed with the QR algorithm to about ten significant figures. Real matrices can have complex eigenvalues, which come in conjugate pairs a ± bi. LU splits a square matrix into a lower and an upper triangular matrix with a row permutation, PA = LU.