Adjoint Matrix Examples for 2×2 and 3×3

Quick answer

Examples follow C_ij = (−1)^(i+j) M_ij, then adj(A) = (C_ij)^T.

Introduction

Reading definitions helps, but adjoints click when you watch numbers move through minors and signs. These worked grids mirror what you will see on homework.

If you need the general method first, read how to find the adjoint matrix and keep the adjoint matrix formula open for notation while you work.

After you study the arithmetic, enter the same matrices in the Adjoint Matrix Calculator. Matching one example fully by hand before you rely on the tool prevents sign errors from hiding in a plausible-looking grid.

What to notice in each example

Zero entries simplify minors because deletion sometimes leaves smaller blocks with zeros already inside.

Sign patterns alternate like a checkerboard. When one cofactor sign is wrong, the transpose still looks plausible, which makes checking important.

Determinant size matters: a 2×2 example may have det(A) = 1, while a nearby 3×3 example can have det(A) = 0 to show singular behavior.

Use the 2×2 shortcut only after you can derive it from cofactors so exam questions on definitions still make sense.

On 3×3 problems, organize cofactors in a table before transposing. Many students lose points by skipping the table.

Example formulas in use

• 2×2: adj([[a,b],[c,d]]) = [[d,−b],[−c,a]]
• 3×3: nine minors with alternating signs
• Verify: A · adj(A) = det(A) · I

Keep fraction arithmetic exact until the final adjoint entries to avoid rounding drift.

Finish each example by computing det(A) separately and verifying A · adj(A).

If the product fails, search for the first cofactor that disagrees with your minor or sign.

Do at least two 3×3 examples fully by hand: one sparse matrix and one dense matrix with mixed signs.

Example walkthroughs

1. Example 1: 2×2 integers. A = [[3, 1], [2, 4]] gives adj(A) = [[4, −1], [−2, 3]] and det(A) = 10.
2. Example 2: 2×2 negative entry. A = [[−1, 2], [5, 0]] gives adj(A) = [[0, −2], [−5, −1]] and det(A) = −10.
3. Example 3: 3×3 sparse. A = [[1, 0, 0], [2, 3, 0], [0, 0, 4]] has cofactors that collapse because of zeros.
4. Example 4: product check. Multiply A by adj(A) for Example 1 and confirm the product equals 10I entry by entry.

Detailed 3×3 sample

Let A = [[2, −1, 0], [0, 1, 3], [1, 0, 2]]. For instance M_11 = det([[1, 3], [0, 2]]) = 2, so C_11 = 2. Continue for all nine positions.

After all C_ij are known, transpose to form adj(A). The process on paper is nine minors, nine signs, one transpose.

Finish by computing det(A) separately and verifying A · adj(A). If det(A) = 0, practice cofactors but do not form A⁻¹.

Compute both a transpose and an adjoint on the same small A in your notebook when vocabulary still feels fuzzy.