Matrix Algebra and Matrix Multiplication

Matrix Operations

Adding and subtracting matrices

In order to add or subtract two matrices, they must be of the same dimension; that is, the two matrices must have the same number of rows and the same number of columns. To add two matrices together, we simply need to add every entry in one matrix to the entry in the same row and same column in the other matrix. For example:

${\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{bmatrix}}={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&a_{13}+b_{13}\\a_{21}+b_{21}&a_{22}+b_{22}&a_{23}+b_{23}\\a_{31}+b_{31}&a_{32}+b_{32}&a_{33}+b_{33}\end{bmatrix}}$

${\begin{bmatrix}1&2&3\\0&1&2\\0&0&1\end{bmatrix}}+{\begin{bmatrix}5&2&1\\3&3&1\\4&2&0\end{bmatrix}}={\begin{bmatrix}1+5&2+2&3+1\\0+3&1+3&2+1\\0+4&0+2&1+0\end{bmatrix}}={\begin{bmatrix}6&4&4\\3&4&3\\4&2&1\end{bmatrix}}$

Multiplying matrices by scalars

When multiplying a matrix by a scalar (or number), all we need to do is multiply each entry of the matrix by the scalar. For example:

$3{\begin{bmatrix}1&2&3\\0&1&2\\0&0&1\end{bmatrix}}={\begin{bmatrix}3&6&9\\0&3&6\\0&0&3\end{bmatrix}}$

$-{\frac {1}{2}}{\begin{bmatrix}2&-4&8\\-2&1&-6\\0&0&7\end{bmatrix}}={\begin{bmatrix}-1&2&-4\\1&-{\frac {1}{2}}&3\\0&0&-{\frac {7}{2}}\end{bmatrix}}$

Multiplying matrices

Matrix multiplication is not commutative; that is, for most matrices $A$ and $B$, $AB\neq BA$.

Resources

• Matrix Addition and Subtraction, Khan Academy
• Multiplying Matrices by Scalars, Khan Academy
• Matrix Multiplication, Khan Academy