# proof of Cramer’s rule

Since $det(A)\ne 0$, by properties of the determinant^{} we know that $A$ is invertible^{}.

We claim that this implies that the equation $Ax=b$ has a unique solution. Note that ${A}^{-1}b$ is a solution since $A({A}^{-1}b)=(A{A}^{-1})b=b$, so we know that a solution exists.

Let $s$ be an arbitrary solution to the equation, so $As=b$. But then $s=({A}^{-1}A)s={A}^{-1}(As)={A}^{-1}b$, so we see that ${A}^{-1}b$ is the only solution.

For each integer $i$, $1\le i\le n$, let ${a}_{i}$ denote the $i$th column of $A$, let ${e}_{i}$ denote the $i$th column of the identity matrix^{} ${I}_{n}$, and let ${X}_{i}$ denote the matrix obtained from ${I}_{n}$ by replacing column $i$ with the column vector^{} $x$.

We know that for any matrices $A,B$ that the $k$th column of the product $AB$ is simply the product of $A$ and the $k$th column of $B$. Also observe that $A{e}_{k}={a}_{k}$ for $k=1,\mathrm{\dots},n$. Thus, by multiplication, we have:

$$\begin{array}{ccc}A{X}_{i}\hfill & =\hfill & A({e}_{1},\mathrm{\dots},{e}_{i-1},x,{e}_{i+1},\mathrm{\dots},{e}_{n})\hfill \\ & =\hfill & (A{e}_{1},\mathrm{\dots},A{e}_{i-1},Ax,A{e}_{i+1},\mathrm{\dots},A{e}_{n})\hfill \\ & =\hfill & ({a}_{1},\mathrm{\dots},{a}_{i-1},b,{a}_{i+1},\mathrm{\dots},{a}_{n})\hfill \\ & =\hfill & {M}_{i}\hfill \end{array}$$ |

Since ${X}_{i}$ is ${I}_{n}$ with column $i$ replaced with $x$, computing the determinant of ${X}_{i}$ with cofactor expansion gives:

$$det({X}_{i})={(-1)}^{(i+i)}{x}_{i}det({I}_{n-1})=1\cdot {x}_{i}\cdot 1={x}_{i}$$ |

Thus by the multiplicative property of the determinant,

$$det({M}_{i})=det(A{X}_{i})=det(A)det({X}_{i})=det(A){x}_{i}$$ |

and so ${x}_{i}=\frac{det({M}_{i})}{det(A)}$ as required.

Title | proof of Cramer’s rule |
---|---|

Canonical name | ProofOfCramersRule |

Date of creation | 2013-03-22 13:03:24 |

Last modified on | 2013-03-22 13:03:24 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 11 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 15A15 |