Add matrices determinants, inverses, special matrices

mechanical/mmme1026_maths_for_engineering.md

@@ -781,3 +781,349 @@ $$
   but $B \ne C$
 
   </details>
+
+## Special Matrices
+
+### Square Matrix
+
+Where $m = n$
+
+<details>
+<summary>
+
+#### Example 1
+
+A $3\times3$ matrix.
+
+</summary>
+
+$$\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}$$
+
+</details>
+
+<details>
+<summary>
+
+#### Example 2
+
+A $2\times2$ matrix.
+
+</summary>
+
+$$\begin{pmatrix}1 & 2 \\ 4 & 5 \end{pmatrix}$$
+
+</details>
+
+### Identity Matrix
+
+The identity matrix is a square matrix whose eleements are all 0, except the leading diagonal which
+is 1s.
+The leading diagonal is the top left to bottom right corner.
+
+It is usually denoted by $I$ or $I_n$.
+
+The identity matrix has the properties that
+
+$$AI = IA = A$$
+
+for any square matrix $A$ of the same order as I, and
+
+$$Ix = x$$
+
+for any vector $x$.
+
+<details>
+<summary>
+
+#### Example 1
+
+The $3\times3$ identity matrix.
+
+</summary>
+
+$$\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$$
+
+</details>
+
+<details>
+<summary>
+
+#### Example 2
+
+The $2\times2$ identity matrix.
+
+</summary>
+
+$$\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}$$
+
+</details>
+
+### Transposed Matrix
+
+The transpose of matrix $A$ of order $m\times n$ is matrix $A^T$ which has the order $n\times m$.
+It is found by reflecting it along the leading diagonal, or interchanging the rows and columns of
+$A$.
+
+![by [Lucas Vieira](https://commons.wikimedia.org/wiki/File:Matrix_transpose.gif)](./images/Matrix_transpose.gif)
+
+Let matrix $D = EF$, then $D^T = (EF)^T = E^TF^T$
+
+#### Example 1
+
+$$
+A = \begin{pmatrix}3 & 2 & 1 \\ 4 & 5 & 6 \end{pmatrix},\,
+A^T = \begin{pmatrix}3 & 4 \\ 2 & 5 \\ 1 & 6\end{pmatrix}
+$$
+
+#### Example 2
+
+$$
+B = \begin{pmatrix}1 \\ 4\end{pmatrix},\,
+B^T = \begin{pmatrix}1 & 4\end{pmatrix}
+$$
+
+#### Example 3
+
+$$
+C = \begin{pmatrix}1 & 2 & 3 \\ 0 & 5 & 1 \\ 2 & 3 & 7\end{pmatrix},\,
+C^T = \begin{pmatrix}1 & 0 & 2 \\ 2 & 5 & 4 \\ 3 & 1 & 7\end{pmatrix}
+$$
+
+### Orthogonal Matrices
+
+A matrix, $A$, such that
+
+$$A^{-1} = A^T$$
+
+is said to be orthogonal.
+
+Another way to say this is
+
+$$AA^T = A^TA = I$$
+
+### Symmetric Matrices
+
+A square matrix which is symmetric about its leading diagonal:
+
+$$A = A^T$$
+
+You can also express this as the matrix $A$, where
+
+$$a_{ij} = a_{ji}$$
+
+is satisfied to all elements.
+
+<details>
+<summary>
+
+#### Example 1
+
+</summary>
+
+$$\begin{pmatrix}
+1 & 0 & -1 & 3 \\
+0 & 3 & 4 & -1 \\
+-2 & 4 & -1 & 6 \\
+3 & -7 & 6 & 2 
+\end{pmatrix}$$
+
+</details>
+
+### Anti-Symmetric
+
+A square matrix is anti-symmetric if
+
+$$A = -A^T$$
+
+This can also be expressed as
+
+$$a_{ij} = -a_{ji}$$
+
+This means that all elements on the leading diagonal must be 0.
+
+<details>
+<summary>
+
+#### Example 1
+
+</summary>
+
+$$\begin{pmatrix}
+0 & -1 & 5 \\
+1 & 0 & 1 \\
+-5 & -1 & 0
+\end{pmatrix}$$
+
+</details>
+
+## The Determinant
+
+### Determinant of a 2x2 System
+
+The determinant of a $2x2$ system is
+
+$$D = a_{11}a_{22} - a_{12}a_{21}$$
+
+It is denoted by
+
+$$
+\begin{vmatrix}
+a_{11} & a_{12} \\
+a_{21} & a_{22}
+\end{vmatrix}
+\text{ or }
+\det
+\begin{pmatrix}
+a_{11} & a_{12} \\
+a_{21} & a_{22}
+\end{pmatrix}
+$$
+
+- A system of equations has a unique solution if $D \ne 0$
+- If $D = 0$, then there are either
+
+  - no solutions (the equations are inconsistent)
+  - intinitely many solutions
+
+### Determinant of a 3x3 System
+
+Let
+
+$$
+A = \begin{pmatrix}
+a_{11} & a_{12} & a_{13} \\
+a_{21} & a_{22} & a_{23} \\
+a_{31} & a_{32} & a_{33}
+\end{pmatrix}
+$$
+
+\begin{align*}
+\det A = &a_{11} \times \det \begin{pmatrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} \\
+        &-a_{12} \times \det \begin{pmatrix}a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} \\
+        &+a_{13} \times \det \begin{pmatrix}a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix}
+\end{align*}
+
+The $2x2$ matrices above are created by removing any elements on the same row or column as its corresponding
+coefficient:
+
+![](./images/vimscrot-2021-11-02T16:19:40,013146580+00:00.png)
+
+### Chessboard Determinant
+
+$\det A$ may be obtained by expanding out any row or column.
+To figure out which coefficients should be subtracted and which ones added use the chessboard
+pattern of signs:
+
+$$\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$$
+
+### Properties of Determinants
+
+- $$\det A = \det A^T$$
+- If all elements of one row of a matrix are multiplied by a constant $z$, the determinant of the
+  new matrix is $z$ times the determinant of the original matrix:
+
+  \begin{align*}
+  \begin{vmatrix} za & zb \\ c & d \end{vmatrix} &= zad - zbc \\
+       &= z(ad-bc) \\
+       &= z\begin{vmatrix} a & b \\ c & d \end{vmatrix}
+  \end{align*}
+
+  This is also true if a column of a matrix is mutiplied by a constant.
+
+  **Application** if the fator $z$ appears in each elements of a row or column of a determinant it
+  can be factored out
+
+  $$\begin{vmatrix}2 & 12 \\ 1 & 3 \end{vmatrix} = 2\begin{vmatrix}1 & 6 \\ 1 & 3 \end{vmatrix} = 2 \times 3
+  \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix}$$
+
+  **Application** if all elements in one row or column of a matrix are zero, the value of the 
+  determinant is 0.
+
+  $$\begin{vmatrix} 0 & 0 \\ c & d \end{vmatrix} = 0\times d - 0\times c = 0$$
+
+
+  **Application** if $A$ is an $n\times n$ matrix,
+
+  $$\det(zA) = z^n\det A$$
+
+- Swapping any two rows or columns of a matrix changes the sign of the determinant
+
+  \begin{align*}
+  \begin{vmatrix} c & d \\ a & b \end{vmatrix} &= cb - ad \\
+        &= -(ad - bc) \\
+        &= -\begin{vmatrix} a & b \\ c & d \end{vmatrix}
+  \end{align*}
+
+  **Application** If any two rows or two columns are identical, the determinant is zero.
+
+  **Application** If any row is a mutiple of another, or a column a multiple of another column, the
+  determinant is zero.
+
+- The value of a determinant is unchanged by adding to any row a constant multiple of another row,
+  or adding to any column a constant multiple of another column
+
+- If $A$ and $B$ are square matrices of the same order then
+
+  $$\det(AB) = \det A \times \det B $$
+
+## Inverse of a Matrix
+
+If $A$ is a square matrix, then its inverse matrix is $A^{-1}$ and is defined by the property that:
+
+$$A^{-1}A = AA^{-1} = I$$
+
+- Not every matrix has an inverse
+- If the inverse exists, then it is very useful for solving systems of equations:
+
+  \begin{align*}
+  A\pmb{x} = \pmb b \rightarrow A^{-1}A\pmb x &= A^{-1}\pmb b \\
+  I\pmb x &= A^{-1}\pmb b \\
+  \pmb x &= A^{-1}\pmb b
+  \end{align*}
+
+  Therefore there must be a unique solution to $A\pmb x = \pmb b$: $\pmb x = A^{-1}\pmb b$.
+
+- If $D = EF$ then 
+
+  $$D^-1 = (EF)^{-1} = F^{-1}E^{-1}$$
+
+### Inverse of a 2x2 Matrix
+
+If $A$ is the $2x2$ matrix
+
+$$
+A = \begin{pmatrix}
+a_{11} & a_{12} \\
+a_{21} & a_{22}
+\end{pmatrix}
+$$
+
+and its determinant, $D$, satisfies $D \ne 0$, $A$ has the inverse $A^{-1}$ given by
+
+$$
+A^{-1} = \frac 1 D \begin{pmatrix}
+a_{22} & -a_{12} \\
+-a_{21} & a_{11}
+\end{pmatrix}
+$$
+
+If $D = 0$, then matrix $A$ has no inverse.
+
+<details>
+<summary>
+
+#### Example 1
+
+Find the inverse of matrix $A = \begin{pmatrix} -1 & 5 \\ 2 & 3 \end{pmatrix}$.
+
+</summary>
+
+1. Calculate the determinant
+
+   $$\det A = -1 \times 3 - 5 \times 2 = -13$$
+
+   Since $\det A \ne 0$, the inverse exists.
+
+2. Calculate $A^{-1}$
+
+   $$ A^{-1} = \frac 1 {-13} \begin{pmatrix} 3 & -5 \\ -2 & -1\end{pmatrix}$$