(New page: * The determinant of any identity matrix is always 1. * If you switch the rows or columns of a matrix, its determinant stays the same. * If you multiply a row (or column) of a matrix (call...) |
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* The determinant of any identity matrix is always 1. | * The determinant of any identity matrix is always 1. | ||
* If you switch the rows or columns of a matrix, its determinant stays the same. | * If you switch the rows or columns of a matrix, its determinant stays the same. | ||
| − | * If you multiply a row (or column) of a matrix (call it A) by a number k, then the determinant of that matrix is k*det(A). For example, if <math>A=\begin{bmatrix} | + | * If you multiply a row (or column) of a matrix (call it A) by a number k, then the determinant of that matrix is k*det(A). For example, if <br> |
| + | <math>A=\begin{bmatrix} | ||
1 & 2 & 3\\ | 1 & 2 & 3\\ | ||
2 & 3 & 5\\ | 2 & 3 & 5\\ | ||
3 & 5 & 0\end{bmatrix}</math> | 3 & 5 & 0\end{bmatrix}</math> | ||
| − | + | <br>then det(A)=8. Thus you can conclude if you multiply the top row by 10 (call the resulting matrix B), then det(B)=10*det(A)=80.<br> | |
| − | then det(A)=8. Thus you can conclude if you multiply the top row by 10 (call the resulting matrix B), then det(B)=10*det(A)=80. | + | <math>\text{det}\begin{pmatrix}10 & 20 & 30\\ |
| − | <math>det\begin{pmatrix}10 & 20 & 30\\ | + | |
2 & 3 & 5\\ | 2 & 3 & 5\\ | ||
3 & 5 & 0\end{pmatrix}=80</math> | 3 & 5 & 0\end{pmatrix}=80</math> | ||
* If you [[transpose]] a matrix (ie turn the rows into columns, and columns into rows), the determinant stays the same. | * If you [[transpose]] a matrix (ie turn the rows into columns, and columns into rows), the determinant stays the same. | ||
| − | * If you multiply matrices and take the determinant, the result is the same as taking the determinants, and then multiplying the determinants. <math>det(A)*det(B)=det(A*B)</math>. | + | * If you multiply matrices and take the determinant, the result is the same as taking the determinants, and then multiplying the determinants. <math>\text{det}(A)*\text{det}(B)=\text{det}(A*B)</math>. |
* It follows that det(A^{-1})=1/det(A). | * It follows that det(A^{-1})=1/det(A). | ||
| − | * It also follows from the above statement that if a matrix <math>B=S^{-1}AS</math>, then <math>det(B)=det(S^{-1}AS)=det(S^{-1})*det(A)*det(S)=det(S)/det(S)*det(A)=det(A).</math> | + | * It also follows from the above statement that if a matrix <math>B=S^{-1}AS</math>, then <br> |
| + | <math>\begin{align}\text{det}(B) & =\text{det}(S^{-1}AS) \\ | ||
| + | & =\text{det}(S^{-1})*\text{det}(A)*\text{det}(S) \\ | ||
| + | & =\text{det}(S)/\text{det}(S)*\text{det}(A) \\ | ||
| + | & =\text{det}(A). \\ \end{align} </math> | ||
[[Category:MA351]] | [[Category:MA351]] | ||
Latest revision as of 09:27, 9 April 2010
- The determinant of any identity matrix is always 1.
- If you switch the rows or columns of a matrix, its determinant stays the same.
- If you multiply a row (or column) of a matrix (call it A) by a number k, then the determinant of that matrix is k*det(A). For example, if
$ A=\begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 5\\ 3 & 5 & 0\end{bmatrix} $
then det(A)=8. Thus you can conclude if you multiply the top row by 10 (call the resulting matrix B), then det(B)=10*det(A)=80.
$ \text{det}\begin{pmatrix}10 & 20 & 30\\ 2 & 3 & 5\\ 3 & 5 & 0\end{pmatrix}=80 $
- If you transpose a matrix (ie turn the rows into columns, and columns into rows), the determinant stays the same.
- If you multiply matrices and take the determinant, the result is the same as taking the determinants, and then multiplying the determinants. $ \text{det}(A)*\text{det}(B)=\text{det}(A*B) $.
- It follows that det(A^{-1})=1/det(A).
- It also follows from the above statement that if a matrix $ B=S^{-1}AS $, then
$ \begin{align}\text{det}(B) & =\text{det}(S^{-1}AS) \\ & =\text{det}(S^{-1})*\text{det}(A)*\text{det}(S) \\ & =\text{det}(S)/\text{det}(S)*\text{det}(A) \\ & =\text{det}(A). \\ \end{align} $
