Matrices

• Matrices can be added or subtracted if they have the same dimensions, and it is performed element -wise.
• Matrices can be multiplied if the number of columns in the first is equal to the number of rows in the second, and it is performed by multiplying each row of the first by each column of the second element -wise, then summed to give an element of the product matrix.
• A zero matrix is where all entries are zero, and multiplying any conformable matrix with it always produces the zero matrix.
• The identity matrix is where the diagonal entries from top left to bottom right are one and all other entries are zero, and multiplying any conformable matrix with it always produces the original matrix.
• The determinant is a function that for a 2x2 matrix: $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$is ad - bc, and for a 3x3 matrix: $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ is a(ei - fh) - b(di - fg) + c(dh - eg), where the bracketed bits are called minors.
• If a matrix has a determinant of zero, it is a singular matrix. Otherwise, it is a non-singular matrix, and the determinant is the scale factor of expansion of the transformation
• To inverse a matrix, do the transpose operation (swap rows and columns) and then find the cofactor matrix (find the minor for each element then sign them in this order: + - + -), then divide by the determinant.
• Singular matrices have no inverse and any matrix multiplied by its inverse equals the identity matrix.
• Linear equations can be expressed in matrix form to find intersections by multiplying the matrix of the linear equation solutions by the inverse of the matrix of the coefficients of the linear equation. (In that specific order, ie (inverse)(soloution) = coordinates)
• For a 3 x 3 system, equations can be interpreted geometrically as a sheaf (singular matrix, infinite solutions), a prism (singular matrix, zero solutions) or a point (non singular matrix).
• Any linear transformation can be represented by a matrix, and the origin is always mapped onto itself.
• Under a transformation, points which are mapped to themselves are known as invariant points, and lines which are mapped to themselves are known as invariant lines.
• For a 2x2 matrix, a reflection in the x - axis is represented by the matrix: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ , and a reflection in the y - axis is represented by the matrix: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ .
• For a 2x2 matrix, a reflection in the line y = x is represented by the matrix: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and a reflection in the line y = - x is represented by the matrix: $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ .
• For a 2x2 matrix, a rotation anticlockwise about the origin through angle $\theta$ is represented by the matrix: $\begin{pmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{pmatrix}$ .
• For a 2x2 matrix, a stretch of scale factor a parallel to the x -axis and of scale factor b parallel to the y -axis is represented by the matrix: $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ .
• For a 3x3 matrix, a reflection in the plane x = 0 is represented by the matrix: $\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ , a reflection in the plane y = 0 is represented by the matrix: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and a reflection in the plane z = 0 is represented by the matrix: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$ .
• For a 3x3 matrix, a rotation anticlockwise by angle $\theta$ about the x axis is represented by the matrix: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & cos\theta & -sin\theta \\ 0 & sin\theta & cos\theta \end{pmatrix}$ , about the y axis is represented by the matrix: $\begin{pmatrix} cos\theta & 0 & sin\theta \\ 0 & 1 & 0 \\ -sin\theta & 0 & cos\theta \end{pmatrix}$ , and about the z axis is represented by the matrix: $\begin{pmatrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$ .
• For successive transformations, PQ represents transformation Q followed by transformation P.
• To find invariant points for a matrix $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ , solve $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ $\begin{pmatrix} x \\ y\end{pmatrix}$ = $\begin{pmatrix} x\\ y\end{pmatrix}$ .
• To find invarient lines for a matrix $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ , solve $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ $\begin{pmatrix} x \\ (mx+c)\end{pmatrix}$ = $\begin{pmatrix} X\\ (mX+c)\end{pmatrix}$ .