Cramer’s Rule

Section 4.3 Cramer’s Rule

Given a matrix equation

A x = b,

where

A

is an

n \times n

matrix. Let

A_{i} (b)

be the matrix obtained from

A

by replacing column

i

by the vector

b .

\begin{array}{llcll} A_{i} (b) = [a_{1} & \dots & b & \dots & a_{n}] \\ ↑ \\ col i \end{array}

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Theorem 4.3.1.

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Let

A

be an invertible

n \times n

matrix. For any

b

R^{n},

the unique solution

x

A x = b

has entries given by

x_{i} = \frac{\det A_{i} (b)}{\det A}, i = 1, 2, \dots, n

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Example 4.3.2.

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Use Cramer’s Rule to solve the system of linear equations.

\begin{aligned} 2 x_{1} - x_{2} & = 5 \\ 3 x_{1} - 5 x_{2} & = 11 \end{aligned}

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