Enter the sixteen dimensionless entries of the real 2×2 blocks A, B, C and D in the partitioned matrix M = [[A, B], [C, D]]. Each entry label gives its row and column within its block. The calculator returns the four entries of S = D − CA⁻¹B, the reduced operator obtained by eliminating the variables associated with A. A must be invertible.

\[\Delta=a_{11}a_{22}-a_{12}a_{21},\qquad A^{-1}=\frac{1}{\Delta}\begin{pmatrix}a_{22}&-a_{12}\\-a_{21}&a_{11}\end{pmatrix},\qquad \Delta\ne0.\]
\[\begin{aligned}T&=A^{-1}B,\\t_{11}&=(a_{22}b_{11}-a_{12}b_{21})/\Delta,\\t_{12}&=(a_{22}b_{12}-a_{12}b_{22})/\Delta,\\t_{21}&=(a_{11}b_{21}-a_{21}b_{11})/\Delta,\\t_{22}&=(a_{11}b_{22}-a_{21}b_{12})/\Delta,\\s_{11}&=d_{11}-(c_{11}t_{11}+c_{12}t_{21}),\\s_{12}&=d_{12}-(c_{11}t_{12}+c_{12}t_{22}),\\s_{21}&=d_{21}-(c_{21}t_{11}+c_{22}t_{21}),\\s_{22}&=d_{22}-(c_{21}t_{12}+c_{22}t_{22}).\end{aligned}\]

For block equations Ax + By = f and Cx + Dy = g, eliminate x using x = A⁻¹(f − By). The remaining equation is Sy = g − CA⁻¹f, where S = D − CA⁻¹B. This is the Schur complement of A, as described in the Stanford block-matrix inversion notes.

Use these four outputs to check block elimination or static condensation. This calculator returns the reduced operator; it does not accept a right-hand side or solve the complete system. Computing S requires invertible A, but does not require invertible D or S.

The defaults are A = [[2, 1], [1, 3]], B = [[1, 2], [3, 4]], C = [[2, 0], [1, 2]] and D = [[5, 1], [0, 6]]. Here det(A) = 5 and A⁻¹B = [[0, 0.4], [1, 1.2]]. Multiplication gives CA⁻¹B = [[0, 0.8], [2, 2.8]], so S = [[5, 0.2], [−2, 3.2]].

A zero computed determinant makes the divisions unavailable, and that unavailable state propagates to all four outputs. An unavailable result is not zero. Values outside the supported numerical range can also make dependent outputs unavailable.

Near-singular A can amplify input and rounding errors, even when the determinant is nonzero. Cancellation in the determinant or final subtraction can further reduce accuracy. The displayed decimal precision does not guarantee that many accurate digits, and the calculator does not estimate conditioning.

This is a fixed calculation for four real 2×2 blocks. It does not perform pivoting, compute pseudoinverses, classify definiteness or support arbitrary matrix sizes.

Reference

  1. Stanford Exploration Project — Inversion of a 2×2 block matrix, equations (18)–(20).