Type a 2×2 or 3×3 matrix, one row per line, and get its determinant, inverse and eigenvalues in one step.
Open in the full calculator → — with every other tool, the AI tutor and PNG export.
From the article Matrix Determinant and Inverse (2×2 and 3×3) by Hand.
Question: Find A−1 for A = [ 4 7 ; 2 6 ] and prove your answer is correct.
Step 1 — determinant: det A = (4)(6) − (7)(2) = 24 − 14 = 10. Non-zero, so the inverse exists.
Step 2 — swap the diagonal, negate the off-diagonal:
adj(A) = [ 6 −7 ; −2 4 ]
Step 3 — divide by the determinant:
A−1 = (1/10) [ 6 −7 ; −2 4 ] = [ 0.6 −0.7 ; −0.2 0.4 ]
Step 4 — multiply back and check you get the identity. Every entry of A · A−1 is a row of A dotted with a column of A−1:
Row 1 × Col 1: (4)(0.6) + (7)(−0.2) = 2.4 − 1.4 = 1
Row 1 × Col 2: (4)(−0.7) + (7)(0.4) = −2.8 + 2.8 = 0
Row 2 × Col 1: (2)(0.6) + (6)(−0.2) = 1.2 − 1.2 = 0
Row 2 × Col 2: (2)(−0.7) + (6)(0.4) = −1.4 + 2.4 = 1
A · A−1 = [ 1 0 ; 0 1 ] = I ✔ The inverse is correct.
Worked in full in Matrix Determinant and Inverse (2×2 and 3×3) by Hand.
This is one of the tools in the Chemistry Calculator Suite, a free calculator built for IIT-JAM, CSIR-NET, GATE and CUET (Chemical Science) preparation. Nothing is behind a login and no result is stored on a server — the arithmetic runs in your own browser.
The Matrix (3×3 / 2×2) calculator above is the live tool — type your values and press its button; the answer appears in the same box. The link under it opens the full suite with this tool already selected, next to the others.