Rules Determinants¶

  1. Determinant of an \(n \times n\) identity matrix is 1. Since the determinant is the product of pivots. In the identity matrix we have only 1’s.

  2. Determinant changes sign when two rows are exchanged.

\[\begin{split}\begin{vmatrix}c & d \\ a & b \end{vmatrix} = - \begin{vmatrix}a & b \\ c & d \end{vmatrix}\end{split}\]

  1. Determinants are linear functions of each row separately: If the first row is multiplied by \(t\), the determinant is multiplied by \(t\). If first rows are added then determinants are added. This rule apply when other rows do not change! $\(\begin{vmatrix}ta & tb \\ c & d\end{vmatrix} = t \begin{vmatrix}a & b \\ c & d \end{vmatrix}\)$

  2. If at least two rows of \(A\) are equal, then \(det(A) = 0\) We go out from rule 2. If we would change 2 identical rows then \(A\) does not change. But row exchanges should swap the signs.

  3. Subtracting a multiple of one row from another row leaves \(det(A)\) unchanged. $\(\begin{vmatrix}a & b \\ c - la & d -lb \end{vmatrix} = \begin{vmatrix}a & b \\ c & d \end{vmatrix}\)$

  4. Matrix with a row of zeros has det \(A\) $\(\begin{vmatrix}0 & 0 \\c &d \end{vmatrix} = 0\)$

    And since we can swap rows the zero row can be anywhere.

  5. If A is triangular then the determinant is the product of diagonal entries $\(\begin{vmatrix}a & b \\0 &d \end{vmatrix} = ad, \space \begin{vmatrix}a & 0 \\c &d \end{vmatrix} = ad\)$

  6. If A is singular then \(det(A) = 0\). If as is invertible then \(det(A) \ne 0\)

  7. The value \(det(AB) = det(A)det(B)\)

\[\begin{split}\begin{vmatrix}a & b \\c &d \end{vmatrix} \begin{vmatrix}p & q \\r &s \end{vmatrix} = \begin{vmatrix}ap + br & aq + bs \\ cp + dr & cq + ds \end{vmatrix}\end{split}\]

  1. The transpose \(A^T\) ha sthe same determinant as A $\(\begin{vmatrix}a & b \\c &d \end{vmatrix} = \begin{vmatrix}a & c \\b &d \end{vmatrix}\)$

Mechanical Energy Expectations