How can the determinant of a 3x3 matrix be computed using minors?

How can the determinant of a 3x3 matrix be computed using minors?

• A. By summing the diagonal products
• B. By multiplying all entries
• C. By computing the determinant of the inverse
• D. By expanding along a row into determinants of 2x2 minors

Answer: (D)

Expanding along a row using minors builds the determinant from smaller sub-determinants. For a 3×3 matrix with first row a, b, c; second row d, e, f; third row g, h, i, the determinant is obtained by taking each entry in the first row, multiplying it by the determinant of the 2×2 matrix you get by removing its row and column, and applying an alternating sign. So you get det = a(ei − fh) − b(di − fg) + c(dh − eg). Here, each ei − fh, di − fg, and dh − eg is the determinant of a 2×2 minor, i.e., the minor associated with the corresponding element. The signs +, −, + come from the position pattern for the row being used. This method, called expansion along a row into 2×2 minors (with cofactors), generalizes to any row or column and is the standard way to compute a 3×3 determinant using minors. The other ideas either describe unrelated operations or are different shortcuts, whereas this approach directly uses the minors.

Expanding along a row using minors builds the determinant from smaller sub-determinants. For a 3×3 matrix with first row a, b, c; second row d, e, f; third row g, h, i, the determinant is obtained by taking each entry in the first row, multiplying it by the determinant of the 2×2 matrix you get by removing its row and column, and applying an alternating sign.

So you get det = a(ei − fh) − b(di − fg) + c(dh − eg). Here, each ei − fh, di − fg, and dh − eg is the determinant of a 2×2 minor, i.e., the minor associated with the corresponding element. The signs +, −, + come from the position pattern for the row being used.

This method, called expansion along a row into 2×2 minors (with cofactors), generalizes to any row or column and is the standard way to compute a 3×3 determinant using minors. The other ideas either describe unrelated operations or are different shortcuts, whereas this approach directly uses the minors.