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# Matrices ## Basic Matrix Operations ### Equality Matrices $A$ and $B$ are equal if they have the same size and $a_{ij} = b_{ij}$ for every $i$ and $j$. ### Addition and Subtraction If $A$ and $B$ are the same size, then $A \pm B$ is the matrix obtained by adding or subtracting the correspondi...

# Matrices ## Basic Matrix Operations ### Equality Matrices $A$ and $B$ are equal if they have the same size and $a_{ij} = b_{ij}$ for every $i$ and $j$. ### Addition and Subtraction If $A$ and $B$ are the same size, then $A \pm B$ is the matrix obtained by adding or subtracting the corresponding entries. ### Scalar Multiplication If $A$ is a matrix and $c$ is a scalar, then $cA$ is the matrix obtained by multiplying each entry of $A$ by $c$. ### Matrix Multiplication If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then $AB$ is an $m \times p$ matrix. The $ij$-th entry of $AB$ is obtained by multiplying the $i$-th row of $A$ by the $j$-th column of $B$. $(AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} +... + a_{in}b_{nj}$ ### Example $ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $ $ A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} $ $ 2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} $ $ AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $ ## Special Matrices ### Zero Matrix A matrix with all entries equal to zero. ### Identity Matrix A square matrix with 1's on the main diagonal and 0's elsewhere. $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ### Transpose The transpose of a matrix $A$, denoted by $A^T$, is obtained by interchanging rows and columns. $(A^T)_{ij} = a_{ji}$ ### Symmetric Matrix A matrix $A$ is symmetric if $A^T = A$. ## Matrix Inversion ### Definition If $A$ is a square matrix, then its inverse, denoted by $A^{-1}$, is a matrix such that $AA^{-1} = A^{-1}A = I$. ### Formula for 2x2 Matrix If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$, provided that $ad - bc \neq 0$. ### Properties of Inverses - $(A^{-1})^{-1} = A$ - $(AB)^{-1} = B^{-1}A^{-1}$ - $(A^T)^{-1} = (A^{-1})^T$