More practice questions

5. Q5. Write the minors and cofactors of the elements of the determinant with rows 1, 0, 0; 0, 1, 0; 0, 0, 1 (NCERT Exercise 4.3, Q2 (i)).

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   Each diagonal element has minor 1 (the 2 × 2 identity is left) and cofactor (−1)²ⁱ × 1 = 1. Every other minor is 0, because the 2 × 2 determinant left has a row of zeros, so every other cofactor is 0 too.

6. Q6. For the determinant of Example 1 (rows 2, −3, 5; 6, 0, 4; 1, 5, −7), check that the elements of column 1 times the cofactors of column 2 add up to 0.

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   Column 1 is 2, 6, 1 and the cofactors of column 2 are A₁₂ = 46, A₂₂ = −19, A₃₂ = 22. So 2(46) + 6(−19) + 1(22) = 92 − 114 + 22 = 0.

7. Q7. A is a 3 × 3 matrix with |A| = 5. Find a₂₁A₂₁ + a₂₂A₂₂ + a₂₃A₂₃ and a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃.

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   The first is row 2 with its own cofactors, which is the determinant: 5. The second is row 2 with the cofactors of row 1, which is 0.

8. Q8. In the determinant with rows 3, 1, 2; 0, k, 4; 1, 0, 5, the cofactor A₁₃ is −1. Find k, and then the value of the determinant.

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   A₁₃ = (−1)^{1 + 3}(0 × 0 − k × 1) = −k, so k = 1. Along row 1: A₁₁ = 5 − 0 = 5 and A₁₂ = −(0 − 4) = 4, so the determinant is 3(5) + 1(4) + 2(−1) = 17.