Definition of a Matrix

A matrix is a list of numbers arranged into rows and columns. They are usually represented with capital letters and they are drawn as arrays of numbers in square brackets. For example, \[ A = \begin{bmatrix} 1 & 3 & -7 \\ 2 & 0 & 0 \\ 3 & -1 & 1 \end{bmatrix} \]

The rows go across and the columns go down. The second row of \(A\) is the vector \(<2, 0, 0>\) and the third column of \(A\) is \[ \begin{bmatrix} -7 \\ 0 \\ 1\end{bmatrix} \]

Specific entries are denoted with subscripts. The element in the first row and second column is \[A_{1,2} = 3\] Keep in mind that the row always comes first and the column second.

The dimensions of a matrix are given as rows \(\times\) columns. So the dimensions of \(A\) are \(2 \times 2.\)

Matrix Addition

If \(A\) and \(B\) are two matrices of the same dimension, then they can be added coordinate-wise.

Example 1: \[ \begin{bmatrix} 2 & 1 & 3 \\ 0 & 4 & 2 \end{bmatrix} + \begin{bmatrix} 7 & -2 & 0 \\ 1 & 1 & 5 \end{bmatrix} = \begin{bmatrix} 2+7 & 1+(-2) & 3+0 \\ 0+1 & 4+1 & 2+5 \end{bmatrix} = \begin{bmatrix} 9 & -1 & 3 \\ 1 & 5 & 7 \end{bmatrix} \] Example 2: \[ \begin{bmatrix} 0 & 5 & 0 \\ -3 & 1 & 1 \end{bmatrix} + \begin{bmatrix} 2 & 8 \\ 0 & -3 \end{bmatrix} \] This sum does not exist the because the first matrix has dimensions \(2 \times 3\) but the second matrix has dimensions \(2 \times 2.\)

Properties of the Matrix Addition

• Commutativity: If \(A\) and \(B\) have the same dimensions, then \[A+B=B+A.\]
• Identity: Given a matrix \(A\) with dimensions \(n \times m,\) there is a zero matrix \(0\) of dimensions \(n \times m\) such that \(A + 0 = 0 + A = 0.\) The zero matrix has \(0\) in every entry.
• Associativity: If \(A,\) \(B\) and \(C\) are matrices that all have the same dimensions, then \((A+B)+C = A+(B+C).\)

Scalar Multiplication

Scalar multiplication works in a similar way for matrices as it does for vectors. When multiplying a matrix by a number, every entry gets multiplied by the number.

Example: Let \[ A = \begin{bmatrix} 2 & 3 \\ 0 & 1 \\ -2 & 4 \end{bmatrix} \] Then \[ 3A = 3 \begin{bmatrix} 2 & 3 \\ 0 & 1 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 3*2 & 3*3 \\ 3*0 & 3*1 \\ 3*-2 & 3*4 \end{bmatrix} = \begin{bmatrix} 6 & 9 \\ 0 & 3 \\ -6 & 12 \end{bmatrix} \]

Matrix Subtraction

Subtracting matrices is a combination of scalar multiplication and matrix addition. If \(A\) and \(B\) are matrices of the same dimensions, \(A - B = A + (-1)B.\)

Example: \[ \begin{bmatrix} 2 & 3 \\ -1 & 0 \\ \end{bmatrix} - \begin{bmatrix} 1 & -2 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ -1 & 0 \\ \end{bmatrix} + \begin{bmatrix} -1 & 2 \\ 0 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 5 \\ -1 & -3 \end{bmatrix} \]