Which method correctly expresses the determinant of a 3x3 matrix by using 2x2 minors?

Which method correctly expresses the determinant of a 3x3 matrix by using 2x2 minors?

• A. Sum of the products of the diagonals
• B. Expansion along a row into determinants of 2x2 minors
• C. Inversion of the matrix
• D. Product of the diagonal entries

Answer: (B)

The determinant of a 3×3 matrix can be computed by expanding along a row into determinants of 2×2 minors. This means you take each entry in a chosen row, multiply it by the determinant of the 2×2 submatrix you get when you remove the row and the column of that entry, and sum with alternating signs. For a matrix with rows [a b c], [d e f], [g h i], the expansion along the first row is det = a*(ei − fh) − b*(di − fg) + c*(dh − eg). Each bracket is a determinant of a 2×2 minor. This is the method described: using 2×2 minors to form the determinant via expansion along a row (or a column). The other options don’t express the determinant in terms of 2×2 minors: the product of diagonals is not generally correct for arbitrary 3×3 matrices, and inversion or a simple sum of diagonal products does not capture the full determinant in general.

The determinant of a 3×3 matrix can be computed by expanding along a row into determinants of 2×2 minors. This means you take each entry in a chosen row, multiply it by the determinant of the 2×2 submatrix you get when you remove the row and the column of that entry, and sum with alternating signs.

For a matrix with rows [a b c], [d e f], [g h i], the expansion along the first row is

det = a*(ei − fh) − b*(di − fg) + c*(dh − eg).

Each bracket is a determinant of a 2×2 minor.

This is the method described: using 2×2 minors to form the determinant via expansion along a row (or a column). The other options don’t express the determinant in terms of 2×2 minors: the product of diagonals is not generally correct for arbitrary 3×3 matrices, and inversion or a simple sum of diagonal products does not capture the full determinant in general.