#### TLDR 1. multiply matrices by matching rows and columns 2. flip them across the diagonal to transpose 3. singular matrices (redundant rows) are not invertible 4. linear systems with an invertible matrices have a unique solution 5. determinants are the change in volume from a linear transformation 6. determinant is negative if it flips, which happens whenever you swap rows 7. determinant is zero when matrix is singular #### Matrix Multiplication To multiply the $m$ x $n$ matrix $A$ by the ${} n$ x ${} p$ matrix $B$, each element of the resulting $m$ x ${} p$ matrix $C$ is the dot product of the corresponding rows of $A$ and the columns of $C$. This can be approached in a few different ways: ![[Pasted image 20251229172211.png|400]] Basic facts: $A(BC) = (AB)C$ $A(B+C)=AB+AC$ $(A+B)C=AC+BC$ $I_{m}A = A + AI_{m}$ #### Transpose Definition: $(A^{\mathrm{T}})_{i,j}=A_{j,i}$ $A$ is symmetric if $A=A^{\mathrm{T}}$ $(AB)^{\mathrm{T}}=B^{\mathrm{T}}A^{\mathrm{T}}$ $(A^{\mathrm{T}})^{\mathrm{T}}=A$ $(A+B)^{\mathrm{T}}=A^{\mathrm{T}}+B^{\mathrm{T}}$ For square, non-singular A we have $(A^{-1})^{\mathrm{T}}=(A^{\mathrm{T}})^{-1}$ #### Inverse If a matrix is square and invertible, there exists a unique inverse $A$ such that: matrices, $AA^{-1}=A^{-1}A=I$ A tall matrix has a left inverse and a right inverse if it is wide. More facts: - The solution to the linear system $Ax=b$ is $x=A^{-1}b$ - $(AB)^{-1}=B^{-1}A^{-1}$ - $\begin{bmatrix}a & b \\ c & d\end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$ To be invertible, a matrix must not be singular (gaussian elimination results in a row of zeros) Let $A$ and $B$ be square matrices and suppose that $A$ is invertible. Then $B$ is invertible if and only if $AB$ is invertible. [[linear systems]] describes gaussian elimination and other methods for computing the inverse to do this Singular matrices: - Noninvertible - $Ax=b$ has no solution for some $b$ #### Determinant ##### Geometric Perspective: The determinant is the signed change in volume of a unit space by some linear transformation $\det(A)= \pm \frac{\text{Volume}(Ax|x \in S)}{\text{Volume}(S)}$ where the sign is negative when the image gets flipped by A. This determinant ratio is constant regardless of $S$. In the 2D case: $\det\begin{bmatrix}a & b \\ c & d\end{bmatrix} = ad-bc$ An $n$ x $n$ matrix $A$ is invertible if and only if $\det(A)\neq 0$ Properties of determinants: - for an $n$ x $n$ matrix $A$, $\det(\lambda A) = \lambda^{n}A$ - for $n$ x $n$ matrices $A$ and $B$, $\det(AB) = \det(A)\det(B)$ - for any invertible matrix $A$, $\det(A^{-1}) = \frac{1}{\det A}$ ##### Algebraic Perspective: $\det(A) = (-1)^{\text{\# of row swaps}} \cdot (\text{product of pivots in ref}(A))$ Row swaps flip the image, therefore need to be accounted for ![[Pasted image 20251229172421.png|400]]