Given the matrices A = \begin{pmatrix} 4 & x \\ -1 & 3 \end{pmatrix} and B = \begin{pmatri… — JAMB UTME

Given the matrices $A = \begin{pmatrix} 4 & x \\ -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , if the determinant of the product matrix $AB$ is $95$ , what is the value of $x$ ?

Cevap: 7

Cevap

The value of

x

7

Using the property

\det(AB) = \det(A)\det(B)

, we find

\det(B) = (2)(4) - (1)(3) = 5

and

\det(A) = (4)(3) - (x)(-1) = 12 + x

. Setting

5(12 + x) = 95

yields

60 + 5x = 95

, which solves to

x = 7

Adım Adım Çözüm

Calculate the determinant of matrix

B

\det(B) = (2)(4) - (1)(3) = 5

The determinant of a

2 \times 2

matrix

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

ad - bc

Express the determinant of matrix

A

in terms of

x

\det(A) = (4)(3) - (x)(-1) = 12 + x

Applying the determinant formula to matrix

A

gives

12 - (-x) = 12 + x

Apply the determinant product property

\det(AB) = \det(A) \cdot \det(B)

5(12 + x) = 95

The determinant of the product of two square matrices is equal to the product of their individual determinants.

Solve the linear equation for

x

60 + 5x = 95 \implies 5x = 35 \implies x = 7

Subtract

60

from both sides and divide by

5

to isolate

x

Anahtar Kavram

Determinant of Matrix Product Property

Soru

Cevap

Adım Adım Çözüm

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