First row of the matrix A is [1 3 2]. If then a possible value of det(A) is

Engineering

Mathematics

Introduction to Determinants

Question

First row of the matrix A is [1 3 2]. If $adj (A) = [\begin{array}{ccc} - 2 & 4 & a \\ - 1 & 2 & 1 \\ 3 a & - 5 & - 2 \end{array}]$ then a possible value of det(A) is

– 1

– 2

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|A| = a₁₁A₁₁ + a₁₂A₂₁ + a₁₃A₃₁

|A| = 1(– 2) + 3(– 1) + 2(3α)

|A| = – 5 + 6α

adj A = [\begin{array}{ccc} - 2 & 4 & α \\ - 1 & 2 & 2 \\ 3 α & - 5 & - 2 \end{array}]

|adj A| = – 2(– 4 + 5) – 4(2 – 3α) + α(5 – 6α)

= – 2 – 8 + 12α + 5α – 6α²

|A|² = – 6α² + 17α – 10

(6α – 5)² = – 6α² + 17α – 10

36α² + 25 – 60α = – 6α³ + 17α – 10

42α² – 77α + 35 = 0

6α² – 11α + 5 = 0

(6α – 5) (α – 1) = 0

α = \frac{5}{6}

or α = 1

|A| = – 5 + 6α

When,

α = \frac{5}{6}, | A | = 0

When, α = 1, |A| = 1

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