A determinant is a single real number computed from a square matrix. For a 2x2 matrix, |a1 b1; a2 b2| = a1b2 - b1a2 (cross-multiply the diagonals and subtract). For a 3x3 matrix, expand along any row or column: multiply each entry of that line by its minor (the smaller determinant left after deleting that entry's row and column), attach the alternating +/- sign pattern, and add. Every row or column yields the same value, which is what makes checking a determinant by re-expanding along a different line possible.
A third-order determinant is evaluated by expanding along a chosen line, multiplying each entry by its minor with the alternating sign pattern.
Expanding along the first row gives 1(−2)−2(16)+(−1)(3)=−37.
Deleting the first row and each column in turn gives the three 2×2 minors, which combine with the +,−,+ signs to the final value.
✓Final answer
13−2201−124=−37.
Expanding along the first row, ∣A∣=10124−23−224+(−1)3−201.
Evaluating each minor: 0124=0−2=−2,3−224=12−(−4)=16,3−201=3−0=3.
So ∣A∣=1(−2)−2(16)+(−1)(3)=−2−32−3=−37.
Verification by expanding along the second column instead (entries 2,0,1, signs −,+,−): ∣A∣=−23−224+0−113−12=−2(16)−1(2−(−3))=−32−5=−37. Both independent expansions agree at −37.
✓Final answer
13−2201−124=−37.
The second column contains a zero (the entry 0 in row 2), so expanding along it - as done in the verification - removes one term and slightly shortens the work.
Students frequently forget the middle minus sign of the +,-,+ pattern and add all three terms. Write the sign explicitly before each term rather than assuming every term is added.