Page 278 M15

Given the matrix \(\small B = \begin{pmatrix} 2-x & 1 \\ 4 & 2-x \end{pmatrix}\) , find all values of x that are solutions of det(B) = 0.


Calculate the determinant of a 2 x 2 matrix using the formula

det(B) = ad - bc

Therefore:
det(B) = (2−x)(2−x)−(1)(4)


Simplify the expression

det(B) = (2−x)(2−x)−(1)(4)
det(B) = (2-x)^2 - 4
det(B)= 4 − 4x + x^2 −4
det(B) = x^2 - 4x


Set to 0 & Factor to solve for X

x(x-4) = 0
x = 0 or x = 4

Therefore:
det(B) = 0 when x = 0 or x = 4. This makes the matrix invertible (singular).