Question 15 of 58intermediate💡 UnderstandMCQ1 mark

If Δ=∣a11a12a13a21a22a23a31a32a33∣\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 Cofactors of a_{ij}, then value of Δ\Delta is given by

Exercise: EXERCISE 4.3 | Q: 5 | (Page 12)
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Explanation

According to the textbook context, the value of a determinant Δ is equal to the sum of the product of elements of any row (or column) with their corresponding cofactors. The context explicitly states: 'Hence Δ = sum of the product of elements of any row (or column) with their corresponding cofactors.' Option (D) shows expansion along column 1, where elements a₁₁, a₂₁, a₃₁ are multiplied with their corresponding cofactors A₁₁, A₂₁, A₃₁ respectively. This is a valid expansion method. Options (A) and (C) are incorrect because when elements of a row are multiplied with cofactors of a different row, the sum equals zero (as stated in the Note). Option (B) is incorrect because it mixes elements and cofactors from different rows and columns without following the proper expansion pattern.

Solution Steps

  1. Step 1: Recall the property that Δ = sum of products of elements of any row (or column) with their corresponding cofactors.

  2. Step 2: Check Option (A): Elements of row 1 (a₁₁, a₁₂, a₁₃) multiplied with cofactors of row 3 (A₃₁, A₃₂, A₃₃) → Sum = 0 (different rows).

  3. Step 3: Check Option (B): Mixed elements and cofactors from different positions → Not a valid expansion.

  4. Step 4: Check Option (C): Elements of row 2 (a₂₁, a₂₂, a₂₃) multiplied with cofactors of row 1 (A₁₁, A₁₂, A₁₃) → Sum = 0 (different rows).

  5. Step 5: Check Option (D): Elements of column 1 (a₁₁, a₂₁, a₃₁) multiplied with their corresponding cofactors (A₁₁, A₂₁, A₃₁) → Valid expansion along column 1, hence equals Δ.

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