A matrix is singular (i.e., it does not have an inverse) if its determinant is zero. To find the value of
x that makes the matrix singular, we need to calculate the determinant of the matrix and set it equal to zero.
The given matrix is:
A=[] To find the determinant of a
3×3 matrix, we use the formula:
det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)where
a,b,c, etc., are the elements of the matrix. Applying this to our matrix, we have:
a=2+x,‌‌b=3,‌‌c=4,d=1,‌‌e=−1,‌‌f=2,g=x,‌‌h=1,‌‌i=−5
Substituting these into the determinant formula, we get:
‌det(A)=(2+x)((−1)(−5)−(2)(1))−3((1)(−5)−(2)(x))+4((1)(1)−(x)(−1))
‌=(2+x)(5−2)−3(−5−2x)+4(1+x)‌=(2+x)(3)+15+6x+4+4x‌=6+3x+15+6x+4+4x‌=25+13x Setting the determinant equal to zero to find when the matrix is singular:
‌25+13x=0‌13x=−25‌x=−‌Thus, the value of
x that makes the matrix singular is
x=−‌.
Checking the options:
Option B correctly states
x=−‌, so Option B is the correct choice.