Given the matrix A = \begin{pmatrix} y & 5 \\ 3 & y+2 \end{pmatrix}, if the determinant of… — JAMB UTME

Given the matrix $A = \begin{pmatrix} y & 5 \\ 3 & y+2 \end{pmatrix}$ , if the determinant of $A$ is $9$ and $y > 0$ , calculate the value of $y$ .

Cevap: 4

The value of

y

is 4.

Expanding the determinant of

A

yields

\det(A) = y(y+2) - 15 = y^2 + 2y - 15

. Equating this to 9 gives

y^2 + 2y - 24 = 0

, which factors as

(y+6)(y-4) = 0

. The roots are

y = -6

and

y = 4

. Given that

y > 0

, the required value is 4.

Find the expression for the determinant of matrix

A

\det(A) = y(y+2) - (5)(3) = y^2 + 2y - 15

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is given by

ad - bc

Form and solve the quadratic equation

y^2 + 2y - 15 = 9 \implies y^2 + 2y - 24 = 0 \implies (y + 6)(y - 4) = 0

Set the determinant expression equal to the given determinant value of 9 and rearrange into standard quadratic form.

Apply the given domain restriction

y > 0

y = 4

The root

y = -6

is discarded because

y

must be strictly positive.

Determinant of a 2x2 matrix and solving non-linear determinant equations

Soru