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Cofactor Expansions

1. Introduction

Introduction

2. Cofactor Expansions

Cofactor Expansions

3. Cramer’s Rule and Matrix Inverses

Cramer’s Rule and Matrix Inverses

4. Definition of a cofactor

Let $A$ be an $n\times n$ matrix and let $A_{ij}$ be the $(i,j)$ minor of $A$ , the matrix obtained by deleting row $i$ and column $j$ of $A$ . Which of the following is the $(i,j)$ cofactor $C_{ij}$ ?

5. Cheapest cofactor expansion

You have to compute the determinant of

A = \left(\begin{array}{ccc} 0 & 2 & 0 \\ 1 & 3 & 4 \\ 5 & 6 & 7 \end{array}\right)

by a single cofactor expansion. Which row or column should you expand along to compute the fewest $2\times 2$ determinants?

6. Expand along a sparse row

Expand the determinant of

A = \left(\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 4 \\ 5 & 6 & 7 \end{array}\right)

along its second row. Enter $\det(A)$ as a number.

7. A three by three determinant

Compute

\det\left(\begin{array}{ccc} 2 & 1 & 3 \\ 0 & -1 & 4 \\ 5 & 2 & 0 \end{array}\right).

Enter a number.

8. Cofactor expansion along a column

For

A = \left(\begin{array}{ccc} 1 & 1 & 4 \\ 1 & 2 & 2 \\ 1 & 2 & 5 \end{array}\right),

expand $\det(A)$ along the third column. Enter the value of $\det(A)$ as a number.

9. In your own words

In your own words, state the single most important definition or theorem of this section, and explain what it says.

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