Cramer’s Rule

Section 3.2 Cramer’s Rule

We will

Develop a theorem that is an algorithm for solving \(A\vec{x}=\vec{b}\text{.}\)
Use this theorem to prove another algorithm about inverses
Decide whether these are useful

Subsection 3.2.1 Cramer’s Rule Illustrated

This activity will show you a relationship between a special determinant and solutions to linear systems.

Let \(M_i(\vec{x})\) be the matrix \(M\) with column \(i\) replaced by \(\vec{x}\text{.}\)

Example 3.2.1.

For \(M=\left[ \begin{array}{rr} 1 & 1 \\ 3 & 2 \end{array} \right]\) and \(\vec{x}=\left[ \begin{array}{r} 8 \\ 5 \end{array} \right]\) we have

\begin{equation*} M_1(\vec{x}) = \left[ \begin{array}{rr} 8 & 1 \\ 5 & 2 \end{array} \right]\text{.} \end{equation*}

Activity 3.2.1.

\begin{equation*} A=\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & 2 & 2 \end{array} \right], \vec{b}=\left[ \begin{array}{r} 2 \\ 9 \\ 11 \end{array} \right], \vec{x}=\left[ \begin{array}{r} x_1 \\ x_2 \\ x_3 \end{array} \right] \end{equation*}

(a)

Write \(A_1(\vec{b})\) and \(I_1(\vec{x})\text{.}\) Note \(I\) is the identiy matrix in this case \(3 \times 3\text{.}\)

(b)

Evaluate \(AI_1(\vec{x})\text{.}\) Compare this result to the previous calculations.

(c)

Calculate \(\det(I_1(\vec{x}))\)

(d)

Simplify (don’t calculate) \(\det(AI_1(\vec{x})))\text{.}\)

(e)

Combine these into a theorem about solving for \(x_i\text{.}\) This is known as Cramer’s Rule.

Subsection 3.2.2 Formula for Inverse

This activity will demonstrate how to use Cramer’s Rule to calculate the inverse of a matrix.

Activity 3.2.2.

This activity considers the solution to \(A\vec{x}=\vec{b}\text{.}\) Let \(A^{-1}=[\vec{o}_1 \ldots \vec{o}_j \ldots \vec{o_n}]\) (each \(\vec{o}_i\) is a column vector).

(a)

What is \(AA^{-1}\text{?}\)

(b)

What is \(A\vec{o}_j\text{?}\)

(c)

Use Cramer’s Rule to find the \(i\)th entry in \(\vec{o}_j\) based on the previous statement.

(d)

Write a formula for the inverse.

Theorem 3.2.2.

\begin{equation*} A^{-1} = \frac{1}{\det A}\left[ \begin{array}{rcrcr} C_{1,1} & \ldots & C_{1,j} & \ldots & C_{1,n} \\ C_{2,1} & \ldots & C_{2,j} & \ldots & C_{2,n} \\ \vdots & & \vdots & & \vdots \\ C_{n,1} & \ldots & C_{n,j} & \ldots & C_{n,n} \end{array} \right]. \end{equation*}

Subsection 3.2.3 Computational Efficiency

In this section we have noted two forumlae. Given we already have a method for calculating both solutions and inverse matrices, we should ask if these new methods are more efficient.

Activity 3.2.3.

(a)

How many times must a matrix be row reduced to solve a system of equations?

Solution.

1 row reduction

(b)

How many times must a matrix be row reduced to solve a system of equations using Cramer’s Rule?

Solution.

\(n\) row reductions: one for each element of the vector

(c)

Does Cramer’s Rule seem practically useful?

(d)

How many times must a matrix be row reduced to find an inverse?

Solution.

One row reduction

(e)

How many times must a matrix be row reduced to find an inverse using the formula?

Solution.

Too many

(f)

Does the formula seem practically useful?

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