Cramer's rule — why must A have non-zero determinant?

Question

Cramer&#039;s rule — why must A have non-zero determinant?

Kaydence Washington · Accepted Answer

If you read just a little farther down on the Wiki entry, it says

x_{i} = \frac{det A_{i}}{det A} .

If

det A = 0

, we're dividing by zero.

sviraju6d · Accepted Answer

More generally, if a matrix has a determinant equal to zero, it is called "singular." This means it is non-invertible, so you cannot compute

x = A^{- 1} b

, because the inverse

A^{- 1}

does not exist.
There are a number of ways to show this, and this fact is actually fundamentally important in linear algebra, but the best way to see is to note that the inverse of a matrix is formed by a permutation of the elements, and the whole matrix is the multiplied by one over the determinant.
For instance

{(\begin{matrix} a & b \\ c & d \end{matrix})}^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})

where

det = a d - b c

Cramer's rule — why must A have non-zero determinant?

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