Question 13 of 58intermediate🔧 ApplyNumerical2 marks

Using Cofactors of elements of second row, evaluate Δ=∣538201123∣\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}.

Correct Answer

7

Exercise: EXERCISE 4.3 | Q: 3 | (Page 12)
For More Understanding

Explanation

This question tests the concept of expanding a determinant using cofactors of a specific row. The student must first find the minors and cofactors of each element in the second row, then apply the expansion formula. The context provides the formula for cofactor calculation: A_{ij} = (-1)^{i+j} M_{ij}, and the expansion rule: Δ = a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}.

Solution Steps

  1. Step 1: Identify elements of second row: a_{21} = 2, a_{22} = 0, a_{23} = 1

  2. Step 2: Find Minor M_{21} = |3 8| = 3×33 \times 3 - 8×28 \times 2 = 9 - 16 = -7 |2 3|

  3. Step 3: Find Cofactor A_{21} = (-1)^{2+1} × M_{21} = (-1) × (-7) = 7

  4. Step 4: Find Minor M_{22} = |5 8| = 5×35 \times 3 - 8×18 \times 1 = 15 - 8 = 7 |1 3|

  5. Step 5: Find Cofactor A_{22} = (-1)^{2+2} × M_{22} = 1×71 \times 7 = 7

  6. Step 6: Find Minor M_{23} = |5 3| = 5×25 \times 2 - 3×13 \times 1 = 10 - 3 = 7 |1 2|

  7. Step 7: Find Cofactor A_{23} = (-1)^{2+3} × M_{23} = (-1) × 7 = -7

  8. Step 8: Expand using second row: Δ = a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} = 2×72 \times 7 + 0×70 \times 7 + 1×(-7) = 14 + 0 - 7 = 7

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