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行列式:定义

1. Determinants

Determinants

2. Introduction

Introduction

3. The Definition of the Determinant

The Definition of the Determinant

4. Magical Properties of the Determinant

Magical Properties of the Determinant

5. Row operations and the determinant

Start with the identity matrix $I_3$ , swap rows 1 and 3, then multiply row 2 by $5$ . Which of the following is the determinant of the matrix you end up with?

6. Spot the singular matrix

Which of the following matrices has determinant $0$ ?

7. Determinant after a row swap

A $4\times 4$ matrix $A$ has $\det(A) = -7$ . The matrix $B$ is obtained from $A$ by swapping two of its rows, and nothing else. What is $\det(B)$ ? Enter a number.

8. Determinant of a triangular matrix

Compute

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

Enter a number.

9. A Vandermonde determinant

Compute the determinant of the Vandermonde matrix

\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{array}\right),

whose rows are $(1,a,a^2)$ for $a=1,2,3$ , using row operations. Enter a number.

10. Determinant of a triangular matrix

Let $A$ be a triangular matrix, with all entries below the diagonal equal to zero. Which formula gives $\det(A)$ ?

11. Triangular determinant

Compute the determinant of the lower triangular matrix

\begin{pmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ 4 & 5 & -2 \end{pmatrix}.

Enter a number.

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