Consider the matrix A= [[-3, x+4, 4],[-12,5x+19,15],[-12,0x+4,x+19]] Find all the values of x that make A singular.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that a singular matrix is a matrix that is not invertible, hence, for A to be singular, its determinant needs to be equal to zero, such that:

`Delta_A = [(-3,x+4,4),(-12,5x+19,15),(-12,4,x+19)]`

`Delta_A = -3(5x + 19)(x + 19) - 12*16 - 12*15(x + 4) + 48(5x + 19) + 12*15 + 12(x + 4)(x + 19)`

`Delta_A = -3(5x^2 + 114x + 361) - 192 - 180x - 720 + 240x + 912 + 180 + 12(x^2 + 23x + 76)`

`Delta_A = -3x^2 - 6x + 9`

You need to solve for x the equation `Delta_A = 0` such that:

`-3x^2 - 6x + 9= 0 => x^2 + 2x - 3 = 0`

Using quadratic formula, yields:

`x_(1,2) = (-2+-sqrt(4 + 12))/2 => x_(1,2) = (-2+-sqrt(16))/2`

`x_(1,2) = (-2+-4)/2 => x_1 = 1; x_2 = -3`

Hence, evaluating the values of x that makes the matrix A singular, yields `x = 1; x = -3.`

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