When evaluating a 3x3 determinant by expanding along the first row, which statement is correct?

When evaluating a 3x3 determinant by expanding along the first row, which statement is correct?

• A. The row or column chosen for expansion is used to form 2x2 minors
• B. The main diagonal is always multiplied together
• C. A 3x3 determinant equals the product of its row sums
• D. Expanding along the first column yields the product of diagonals

Answer: (A)

Expansion along a row or column uses 2x2 minors. When you pick a row (like the first row) to expand, you take each entry in that row, multiply it by the determinant of the 2x2 submatrix you get by removing that entry’s row and column (the minor), and apply a plus/minus pattern: plus for the first and third positions and minus for the middle in the first row. So the determinant is a times the first 2x2 minor, minus b times the second 2x2 minor, plus c times the third 2x2 minor. For a concrete 3x3 matrix |a b c| |d e f| |g h i| the minors are: - for a: det[[e f],[h i]] = ei − fh - for b: det[[d f],[g i]] = di − fg - for c: det[[d e],[g h]] = dh − eg Therefore the determinant is a(ei − fh) − b(di − fg) + c(dh − eg). This illustrates the mechanism: you form 2x2 minors from the chosen row (or column) and combine them with the corresponding row (or column) entries and signs. The other statements don’t describe the general method. The determinant is not simply the product of the main diagonals, except in special cases like triangular matrices. It also isn’t the product of row sums, and expanding along a column yields a sum of terms involving 2x2 minors with signs, not a product of diagonals.

Expansion along a row or column uses 2x2 minors. When you pick a row (like the first row) to expand, you take each entry in that row, multiply it by the determinant of the 2x2 submatrix you get by removing that entry’s row and column (the minor), and apply a plus/minus pattern: plus for the first and third positions and minus for the middle in the first row. So the determinant is a times the first 2x2 minor, minus b times the second 2x2 minor, plus c times the third 2x2 minor.

For a concrete 3x3 matrix

|a b c|

|d e f|

|g h i|

the minors are:

• for a: det[[e f],[h i]] = ei − fh

• for b: det[[d f],[g i]] = di − fg

• for c: det[[d e],[g h]] = dh − eg

Therefore the determinant is a(ei − fh) − b(di − fg) + c(dh − eg). This illustrates the mechanism: you form 2x2 minors from the chosen row (or column) and combine them with the corresponding row (or column) entries and signs.

The other statements don’t describe the general method. The determinant is not simply the product of the main diagonals, except in special cases like triangular matrices. It also isn’t the product of row sums, and expanding along a column yields a sum of terms involving 2x2 minors with signs, not a product of diagonals.