For how many values (s) of x  in the closed interval [−4, −1],  is the matrix 3−1+x23−1x+2x+3−12  singular?

The given matrix is singular, if  3−1+x23−1x+2x+3−12=0 Applying R2→R2−R1,  R3→R3−R 1Applying C1→C1+C2+C3x+4−1+x20−xx−−x0=0⇒                      x2(x+4)=0⇒                            x=0, −4Hence, x=−4  is the only solution which lies in the given interval.