Determinants

The determinant is zero when the matrix has no inverse.

$$\begin{vmatrix}a & b \ c & d \end{vmatrix}=ad-bc$$

The Properties of Determinants

  1. The determinant of the $n$ by $n$ identity matrix is 1

$$\begin{vmatrix}1 & 0 \ 0 & 1 \end{vmatrix}=1
\text{ and }
\begin{vmatrix}1 & \ & \ddots & \ & & 1 \end{vmatrix}=1$$

  1. The determinant changes sign when two rows are exchanged

$$\begin{vmatrix}c & d \ a & b \end{vmatrix}=-\begin{vmatrix}c & d \ a & b \end{vmatrix}$$

  1. The determinant is a linear function of each row separately

$$
\begin{array}{l}
\text{multiply row 1 by any number }t &
\begin{vmatrix} ta & tb \ c & d \end{vmatrix} =
t\begin{vmatrix} a & b \ c & d \end{vmatrix} \
\text{add row 1 of }A\text{ to row 1 of }A’: &
\begin{vmatrix} a + a’ & b + b’ \ c & d \end{vmatrix} =
\begin{vmatrix} a & b \ c & d \end{vmatrix} +
\begin{vmatrix} a’ & b’ \ c & d \end{vmatrix}
\end{array}
$$

  1. If two rows of A are equal, then $det A = 0$

  2. Subtracting a multiple of one row from another row leaves $det A$ unchanged.

$$\begin{vmatrix}a & b \ c-la & d-la \end{vmatrix}=\begin{vmatrix}a & b \ c & d \end{vmatrix}$$

  1. A matrix with a row of zeros has $det A = 0$

  2. If $A$ is triangular then $det A = a{11}a{22} \cdots a{nn} =$ product ofdiagonal entries.

  3. If $A$ is singular then $det A = 0$. If $A$ is invertible then $det A \neq 0$

  4. The determinant of $AB$ is $det A$ times $det B$ : $|AB| = |A||B|$

  5. The transpose $A^T$ has the same determinant as $A$.

Cramer’s Rule

Cramer’s Rule If $det A$ is not zero , $Ax=b$ is solved by determinants:

$$
x_1 = \frac {det B_1} {det A}
\qquad
x_2 = \frac {det B_2} {det A}
\quad \ldots \quad
x_n = \frac {det B_n} {det A}
$$

The matrix $B_j$ has the $j$th column of $A$ replaced by the vector $b$

example:

转载请注明：转载自srzyhead的博客(https://srzyhead.github.io)

本文链接地址: 线性代数导论 (5-Determinants)