The matrix [x−1−3x+1−2] \left[\begin{array}{cc} x-1&-3 \ x+1&-2 \end{array} ight] will NOT have an inverse where:

Question

The matrix [x−1−3x+1−2] \left[\begin{array}{cc} x-1&-3 \ x+1&-2 \end{array}ight] will NOT have an inverse where:

BlackTom AI · Accepted Answer

{
  "reasoning": "To determine when the matrix is non-invertible, we look at its determinant, since a matrix is invertible iff its determinant is nonzero.
The given matrix is [[x-1, -3], [x+1, -2]]. Its determinant is det = (x-1)(-2) - (-3)(x+1).
Compute step by step: (x-1)(-2) = -2x + 2, and (-3)(x+1) = -3x - 3. Subtracting these gives det = (-2x + 2) - (-3x - 3) = -2x + 2 + 3x + 3 = x + 5.
For the matrix to not have an inverse, the determinant must be zero. Setting det = 0 yields x + 5 = 0, so x = -5.
Thus, the matrix will NOT have an inverse at x = -5. Since the option provided is "b. x=-5", this is the specific value that makes the determinant zero and the matrix singular.
If x ≠ -5, the determinant x + 5 ≠ 0, and the matrix is invertible. The key idea is that determinant zero signals non-invertibility, while a nonzero determinant guarantees an inverse."
}

The matrix [x−1−3x+1−2] \left[\begin{array}{cc} x-1&-3 \\ x+1&-2 \end{array}\right] will NOT have an inverse where:单项选择题

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