The matrix [math: [234x+1]] \left[\begin{array}{cc} 2&3 \ 4&x+1 \end{array} ight] will NOT have an inverse where:

Question

The matrix [math: [234x+1]] \left[\begin{array}{cc} 2&3 \ 4&x+1 \end{array}ight] will NOT have an inverse where:

BlackTom AI · Accepted Answer

Let's restate the problem in our own words: We are given the 2x2 matrix with entries 2, 3 in the first row and 4, x+1 in the second row, and we are asked for which value(s) of x the matrix will NOT have an inverse.

Option a: x = 6. If we compute the determinant, a matrix is invertible exactly when its determinant is not zero. Here det = 2*(x+1) - 3*4 = 2x + 2 - 12 = 2x - 10 = 2(x - 5). Substituting x = 6 gives det = 2*(6) - 10 = 12 - 10 = 2, which is nonzero. Therefore, with x = 6 the matrix does have an inverse, so this value does not make it non-invertible.

Option b: x = 3. Using the determinant formula det = 2x - 10, substitute x = 3 to get det = 2*3 - 10 = 6 - 10 = -4, which is nonzero. Since the determinant is nonzero, the matrix is invertible. Thus, x = 3 does not cause a lack of an inverse.

Option c: x = 5. Plugging x = 5 into the determinant det = 2x - 10 yields det = 2*5 - 10 = 10 - 10 = 0. A determinant of zero means the matrix is singular and does not have an inverse. This is exactly the condition we’re looking for: the value of x that prevents an inverse.

Option d: x = 4. For x = 4, det = 2*4 - 10 = 8 - 10 = -2, which is nonzero. A nonzero determinant confirms the existence of an inverse, so this value does not cause non-invertibility.

In summary, the determinant is a linear function of x, det = 2x - 10, which equals zero when x = 5 and nonzero for all other x. The only value among the options that yields a zero determinant—and thus no inverse—is x = 5, corresponding to option c.

The matrix [math: [234x+1]] \left[\begin{array}{cc} 2&3 \\ 4&x+1 \end{array}\right] will NOT have an inverse where:

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