If [ array ccc 2+x & 3 & 4 1 & -1 & 2 x & 1 & -5 array ] is a singular matrix, then x is

Step-by-step solution

A matrix is singular if its determinant is equal to zero. Let A = bmatrix 2+x & 3 & 4 1 & -1 & 2 x & 1 & -5 bmatrix . The determinant |A| is given by: |A| = (2+x) vmatrix -1 & 2 1 & -5 vmatrix - 3 vmatrix 1 & 2 x & -5 vmatrix + 4 vmatrix 1 & -1 x & 1 vmatrix = 0 Calculating the 2 2 determinants: |A| = (2+x)(5 - 2) - 3(-5 - 2x) + 4(1 - (-x)) = 0 |A| = (2+x)(3) - 3(-5 - 2x) + 4(1 + x) = 0 Expanding the terms: 6 + 3x + 15 + 6x + 4 + 4x = 0 Combining like terms: 13x + 25 = 0 13x = -25 x = - 25 13 Answer: - 25 13