Multiplication and Inverse Matrices

We have $AB=C$. $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then $C$ is an $m\times p$ matrix. We use $c_{ij}$ to denote the entry in row $i$ and column $j$ of matrix $C$ and the same denotation applies to $a_{ij}$ and $b_{ij}$.

Row times column §

$c_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$

Columns §

The product of matrix $A$ and column $j$ of matrix $B$ equals column $j$ of matrix $C$. This tells us that the columns of $C$ are combinations of columns of $A$.

$$A\left[\begin{array}{ccc}|& |& |\\ column1& column2& column3\\ |& |& |\end{array}\right]=\left[\begin{array}{ccc}|& |& |\\ A(column1)& A(column2)& A(column3)\\ |& |& |\end{array}\right]$$

Rows §

The product of row $i$ of matrix $A$ and matrix $B$ equals row $i$ of matrix $C$. So the rows of $C$ are combinations of rows of $B$.

$$\left[\begin{array}{ccc}---& row1& ---\\ ---& row2& ---\\ ---& row3& ---\end{array}\right]B=\left[\begin{array}{ccc}---& (row1)B& ---\\ ---& (row2)B& ---\\ ---& (row3)B& ---\end{array}\right]$$

Column times row §

$$AB=\sum _{k=1}n\left[\begin{array}{c}{a}_{1k}\\ ⋮\\ a_{mk}\end{array}\right]\left[\begin{array}{ccc}{b}_{k1}& \cdots & b_{kp}\end{array}\right]$$

Blocks §

$$\left[\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right]\left[\begin{array}{cc}{B}_1& B_2\\ B_3& B_4\end{array}\right]=\left[\begin{array}{cc}{A}_1{B}_1+A_2{B}_3& A_1{B}_2+A_2{B}_4\\ A_3{B}_1+A_4{B}_3& A_3{B}_2+A_4{B}_4\end{array}\right]$$

Inverses §

If $A$ is singular or not invertible,

then A does not have an inverse,

and we can find some non-zero vector $\mathbf{x}$ for which $A\mathbf{x}=0$

Gauss-Jordan Elimination §

$$E\left[\begin{array}{cc}A& I\end{array}\right]=\left[\begin{array}{cc}I& E\end{array}\right]$$

If $EA=I$, then $E=A^{-1}$.