Given that x is a positive real number and the determinant of the matrix A = \begin{pmatri… — JAMB UTME

Given that $x$ is a positive real number and the determinant of the matrix $A = \begin{pmatrix} x & 1 & 0 \\ 2 & x & 3 \\ 0 & 1 & 4 \end{pmatrix}$ is $2$ , find the value of $x$ .

Cevap: 2

Cevap

The positive value of

x

2

Expanding the determinant along the first row yields

\det(A) = x(4x - 3) - 1(8 - 0) = 4x^2 - 3x - 8

. Equating this to

2

gives

4x^2 - 3x - 10 = 0

. Factoring the quadratic gives

(4x + 5)(x - 2) = 0

, which yields

x = 2

x = -1.25

. Because

x

must be positive, the correct value is

2

Adım Adım Çözüm

Expand the

3 \times 3

determinant along the first row

\det(A) = x(4x - 3) - 1(8 - 0) + 0 = 4x^2 - 3x - 8

Expanding along the first row leverages the zero entry to simplify calculation of the determinant.

Equate the determinant expression to the given determinant value

4x^2 - 3x - 10 = 0

Setting the calculated determinant equal to

2

creates a quadratic equation in terms of

x

Factor the quadratic equation to find the candidate values for

x

(4x + 5)(x - 2) = 0 \implies x = 2 \text{ or } x = -1.25

Factoring determines all algebraic solutions that satisfy the determinant equation.

Apply the positivity constraint given in the problem statement

x = 2

The question restricts

x

to positive real numbers, discarding the negative root.

Anahtar Kavram

Determinant of a 3x3 matrix and quadratic equations

Tahmini Süre:2m 0s

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Adım Adım Çözüm

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