Final answer:
For a matrix to be invertible, its determinant must be non-zero. Calculating the determinant, which depends on the matrix's size and elements, allows you to identify the values of 'x' for which the matrix can or cannot be inverted. Any 'x' that results in a zero determinant makes the matrix non-invertible.
Step-by-step explanation:
To determine for what values of x a matrix is invertible, we need to find when the matrix has a non-zero determinant. A square matrix is invertible if and only if its determinant is not equal to zero because the determinant measures the scaling factor of the linear transformation described by the matrix, and a zero determinant indicates that the matrix maps the n-dimensional space into a space of lower dimension, thus being non-invertible.
The determinant of a 2x2 matrix, for example, which is of the form A = [[a, b], [c, d]], is calculated as ad - bc. For the matrix to be invertible, ad - bc ≠ 0. Similarly, for a 3x3 matrix with elements a_ij, the determinant is a bit more complex to calculate, but the same principle applies: the determinant must be non-zero.
To find the determinant of a given matrix, apply the determinant formula specific to its dimensions or use row reductions to convert it to an upper triangular form, where the determinant is the product of the diagonal elements. Once again, these diagonal elements product must be non-zero for the matrix to be invertible.
For instance, if you are given a matrix with a variable x and are asked to determine for what values of x the matrix is invertible, you would calculate the determinant and set it equal to zero and solve for x. The values of x that make the determinant zero are the ones for which the matrix is not invertible, and all other values will make the matrix invertible.