Δ = det[a₁₁, a₁₂, a₁₃; a₂₁, a₂₂, a₂₃; a₃₁, a₃₂, a₃₃] and Aᵢⱼ is the cofactor of aᵢⱼ. Which expression equals Δ?

a₁₂A₁₂ + a₂₂A₂₂ + a₃₂A₃₂

a₁₂A₂₁ + a₂₂A₂₂ + a₃₂A₂₃

a₁₂M₁₂ + a₂₂M₂₂ + a₃₂M₃₂

a₁₁A₁₂ + a₂₁A₂₂ + a₃₁A₃₂

Question 41

MediumOriginal question

$\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ and $A_{ij}$ is the cofactor of $a_{ij}$ . Which expression equals $\Delta$ ?

a_{12}A_{21} + a_{22}A_{22} + a_{32}A_{23}

Correct answerYour answer

a_{12}M_{12} + a_{22}M_{22} + a_{32}M_{32}

Correct answerYour answer

a_{11}A_{12} + a_{21}A_{22} + a_{31}A_{32}

Correct answerYour answer

a_{12}A_{12} + a_{22}A_{22} + a_{32}A_{32}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

A determinant is the sum of the elements of any one row or column, each times its own cofactor.
The first expression does this for column 2. The minors version drops the signs, the swapped indices pair elements with the wrong cofactors, and column 1's elements with column 2's cofactors give 0.

Ten questions from Determinants, Adjoint & Inverse, at your own pace.