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A matrix is a rectangular array of elements:

{$$\large \begin{bmatrix}23 & 11 &-13\\12 & \sqrt 5 & 3\end{bmatrix} $$}

As can be seen from the example, the elements are often numbers, and could belong to the real or to the complex numbers. However, other mathematical objects could be elements in matrices, such as other matrices.

The positions of numbers in a matrix are significant. Two matrices are not the same, if they contain the same number in a different order.

The mnemonic for matrices is: "Matrices are Roman Catholic." The dimension of a matrix is expressed by the number of Rows it contains by the number of Columns it contains. Likewise, positions in the matrix are often represented by subscripts that give the Row and the Column number of the element. Matrices are often represented by boldface Latin capital letters.

{$$\Large \mathbf A = \begin{bmatrix}a_{\color{red}{1},\color{blue}{1}} & a_{\color{red}{1},\color{blue}{2}} &a_{\color{red}{1},\color{blue}{3}} \\ a_{\color{red}{2},\color{blue}{1}} & a_{\color{red}{2},\color{blue}{2}} & a_{\color{red}{2},\color{blue}{3}}\end{bmatrix} $$}

Matrices are equal if they have the same dimensions and every element of one is equal to the corresponding element of the other.

Matrices of the same dimension can be added by adding their corresponding elements.

{$$\large \begin{bmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ a_{3,1} & a_{3,2} \end{bmatrix} + \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \\ b_{3,1} & b_{3,2} \end{bmatrix} = \begin{bmatrix} a_{1,1}+b_{1,1} & a_{1,2}+b_{1,2} \\ a_{2,1}+b_{2,1} & a_{2,2}+b_{2,2} \\ a_{3,1}+b_{3,1} & a_{3,2}+b_{3,2} \end{bmatrix} $$}

The additive identity of a matrix is the a matrix of the same dimensions filled with zeros.

{$$\large \begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2} & a_{2,3}\\ a_{3,1} & a_{3,2} & a_{3,3} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} =\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2} & a_{2,3}\\ a_{3,1} & a_{3,2} & a_{3,3} \end{bmatrix}$$}

Matrices may be multiplied by a scalar by multiplying every element of the matrix by the scalar.

{$$\Large k\mathbf A = \begin{bmatrix}ka_{1,1} & ka_{1,2} &ka_{1,3} \\ ka_{2,1} & ka_{2,2} & ka_{2,3}\end{bmatrix} $$}

The transform of a matrix is the matrix with the rows and columns interchanged.

{$$\large \mathbf A = \begin{bmatrix}23 & 11 &-13\\12 & \sqrt 5 & 3\end{bmatrix} \quad \mathbf A^{\mathsf T} = \begin{bmatrix}23 & 12 \\11 & \sqrt 5 \\-13 & 3 \end{bmatrix} $$}

Matrix multiplication is defined when the number of columns in the left matrix is equal to the number of rows in the right matrix.

{$$ \Large \mathbf A_{m\times n} \mathbf B_{n\times p} = \mathbf C_{m\times p} $$}

Matrix multiplication in general is ** not** commutative even when both products are defined.

{$$ \Large \mathbf {AB} \ne \mathbf {BA} $$}

Matrices are multiplied by summing the products of the elements in a row of the left matrix and the elements of the corresponding column in the left matrix to produce each element in resulting matrix.

{$$ \large \begin{bmatrix} \color{red}{3} & \color{red}{2} & \color{red}{1} \\ 4 & 0 & 7 \\ -2 & 5 & 3 \end{bmatrix} \times \begin{bmatrix} \color{red}{2} & 5 \\ \color{red}{1} & 6 \\ \color{red}{0} & 3 \end{bmatrix} = \begin{bmatrix} \color{red}{8} & 30 \\ 8 & 41 \\ 1 & 29 \end{bmatrix} $$}

The 1,1 element of the product matrix is computed by summing the products of the elements in the first row of the left matrix and the elements of the first column of the right matrix:

3(2)+2(1)+1(0) = 8

{$$ \large \begin{bmatrix} \color{red}{3} & \color{red}{2} & \color{red}{1} \\ 4 & 0 & 7 \\ -2 & 5 & 3 \end{bmatrix} \times \begin{bmatrix} 2 & \color{red}{5} \\ 1 & \color{red}{6} \\ 0 & \color{red}{3} \end{bmatrix} = \begin{bmatrix} 8 & \color{red}{30} \\ 8 & 41 \\ 1 & 29 \end{bmatrix} $$}

The 1,2 element of the product matrix is computed by summing the products of the elements in the first row of the left matrix and the elements of the second column of the right matrix:

3(5)+2(6)+1(3) = 30

{$$ \large \begin{bmatrix} 3 & 2 & 1 \\ \color{blue}4 & \color{blue}0 & \color{blue}7 \\ -2 & 5 & 3 \end{bmatrix} \times \begin{bmatrix} \color{blue}{2} & 5 \\ \color{blue}{1} & 6 \\ \color{blue}{0} & 3 \end{bmatrix} = \begin{bmatrix} 8 & 30 \\ \color{blue}{8} & 41 \\ 1 & 29 \end{bmatrix} $$}

The 2,1 element of the product matrix is computed by summing the products of the elements in the second row of the left matrix and the elements of the first column of the right matrix:

4(2)+0(1)+1(0) = 8

and so forth.

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