Section01 The Properties of Determinants

The Properties of the Determinants

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

   $$\left\vert \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right\vert = 1 \text{ and } \left\vert \begin{matrix} 1 & & \\ & \ddots & \\ & & 1 \end{matrix} \right\vert = 1$$

2. The determinant changes sign when two rows are exchanged. (sign reversal)

   $$\left\vert \begin{matrix} c & d \\ a & b \end{matrix} \right\vert = -\left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert$$

3. The determinant is a linear function of each row separately. (all other rows stay fixed.)

   • Multiply row 1 by any number $t$ $\det$ is multiplied by $t$ $$\left\vert \begin{matrix} ta & tb \\ c & d \end{matrix} \right\vert = t \left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert$$
   • Add row 1 of $\boldsymbol{A}$ to row 1 of $\boldsymbol{A}'$: then determinant add $$\left\vert \begin{matrix} a + a' & b + b' \\ c & d \end{matrix} \right] = \left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert + \left\vert \begin{matrix} a' & b' \\ c & d \end{matrix} \right\vert$$

4. If two rows of $\boldsymbol{A}$ are equal, then $\det \boldsymbol{A} = 0$

   $$\left\vert \begin{matrix} a & b \\ a & b \end{matrix} \right\vert = 0$$

5. Subtracting a multiple of one row from another row leaves $\det \boldsymbol{A}$unchanged.

   $$\left\vert \begin{matrix} a & b \\ c - \ell a & d - \ell b \end{matrix} \right\vert = \left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert$$

   • From rule 3

     $$\left\vert \begin{matrix} a & b \\ c - \ell a & d - \ell b \end{matrix} \right\vert = \left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert - \left\vert \begin{matrix} a & b \\ \ell a & \ell b \end{matrix} \right\vert = \left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert$$

6. A matrix with a row of zeros has $\det \boldsymbol{A} = 0$

   $$\left\vert \begin{matrix} 0 & 0 \\ c & d \end{matrix} \right\vert = 0 \text{ and } \left\vert \begin{matrix} a & b \\ 0 & 0 \end{matrix} \right\vert = 0$$

7. If $\boldsymbol{A}$ is triangular then $\det \boldsymbol{A} = a_{11}a_{22}\cdots a_{nn} = \text{product of diagonal entries.}$

   $$\left\vert \begin{matrix} a & b \\ 0 & d \end{matrix} \right\vert = ad \text{ and } \left\vert \begin{matrix} a & 0 \\ c & d \end{matrix} \right\vert = ad$$

8. If $\boldsymbol{A}$ is singular then $\det \boldsymbol{A} = 0$. If $\boldsymbol{A}$ is invertible then $\det \boldsymbol{A} \ne 0$.

9. The determinant of $\boldsymbol{AB}$ is $\det \boldsymbol{A}$ times $\det \boldsymbol{B}$: $\vert \boldsymbol{AB} \vert = \vert \boldsymbol{A} \vert \vert \boldsymbol{B} \vert$

   $$\left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert \left\vert \begin{matrix} p & q \\ r & s \end{matrix} \right\vert = \left\vert \begin{matrix} ap + br & aq + bs \\ cp + dr & cq + ds \end{matrix} \right\vert$$ $$\det \boldsymbol{A} \times \det \boldsymbol{A}^{-1} = \det (\boldsymbol{A}\boldsymbol{A}^{-1}) = \det \boldsymbol{I} = 1$$

10. The transpose $\boldsymbol{A}^{\text{T}}$ has the same determinant as $\boldsymbol{A}$

    $$\left\vert \begin{matrix} a & b \\ c & d \end{matrix} \right\vert = \left\vert \begin{matrix} a & c \\ b & d \end{matrix} \right\vert$$

Calculating in Matlab

$$\begin{array}{c} \det \boldsymbol{A} = \pm \det \boldsymbol{U} = \pm (\text{Product of the pivots.}) \\ \pm \text{ is determinated by the times of row exchange.} \end{array}$$

Properties from Exercises

1. $\color{#D0104C}{(\det \boldsymbol{A} + \boldsymbol{B}) \ne \det \boldsymbol{A} + \det \boldsymbol{B}!!!!!}$
2. For every orthogonal matrix ($\boldsymbol{Q}^{\text{T}}\boldsymbol{Q} = \boldsymbol{I}$), $\det Q = \pm 1$
   • proof $$\begin{array}{ll} \because & (\det \boldsymbol{Q}^{\text{T}})(\det \boldsymbol{Q}) = (\det \boldsymbol{Q}^{\text{T}}\boldsymbol{Q}) \text{ and } \det \boldsymbol{Q}^{\text{T}} = \det \boldsymbol{Q}\\ \therefore & (\det \boldsymbol{Q}^{\text{T}})(\det \boldsymbol{Q}) = (\det \boldsymbol{Q})^{2} = (\det \boldsymbol{I}) = 1 \\ \therefore & \det \boldsymbol{Q} = \pm 1 \\ \end{array}$$
3. $(\det c \boldsymbol{A}) = c^{n} \cdot \det \boldsymbol{A}$
4. skew-symmetric matrix ($\boldsymbol{A}^{\text{T}} = - \boldsymbol{A}$); $\det \boldsymbol{A} = 0$ (when the size of $\boldsymbol{A}$ is odd.