Matrices: Addition and Subtraction

Introduction to Matrices

Matrices are a fundamental tool in mathematics, used extensively in solving systems of linear equations.

Definition

A matrix is a rectangular array of numbers.

Dimensions of a Matrix

When examining a matrix, the first consideration should be its dimensions, expressed as rows by columns.

  • Example:

    A matrix with two rows and three columns is referred to as a "two by three" matrix.

Matrix Addition

Matrices can be added together if and only if they have the exact same dimensions.

  • Process:
    Addition is performed by summing corresponding elements in the matrices.

  • Example:

    Given two 2x3 matrices:
    [5amp;7amp;2 −1amp;0amp;3]+[1amp;4amp;−2 6amp;1amp;8]\begin{bmatrix} 5 & 7 & 2 \ -1 & 0 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 4 & -2 \ 6 & 1 & 8 \end{bmatrix}

    The sum is calculated as:
    [5+1amp;7+4amp;2+(−2) −1+6amp;0+1amp;3+8]=[6amp;11amp;0 5amp;1amp;11]\begin{bmatrix} 5+1 & 7+4 & 2+(-2) \ -1+6 & 0+1 & 3+8 \end{bmatrix} = \begin{bmatrix} 6 & 11 & 0 \ 5 & 1 & 11 \end{bmatrix}

Matrix Subtraction

Similar to addition, matrices can be subtracted only if they possess identical dimensions.

  • Process:
    Subtraction involves subtracting corresponding elements in the matrices.

  • Example:

    Consider two 2x3 matrices:
    [5amp;7amp;2 −1amp;0amp;3]−[1amp;4amp;−2 6amp;1amp;8]\begin{bmatrix} 5 & 7 & 2 \ -1 & 0 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 4 & -2 \ 6 & 1 & 8 \end{bmatrix}

    The difference is:

    [5−1amp;7−4amp;2−(−2) −1−6amp;0−1amp;3−8]=[4amp;3amp;4 −7amp;−1amp;−5]\begin{bmatrix} 5-1 & 7-4 & 2-(-2) \ -1-6 & 0-1 & 3-8 \end{bmatrix} = \begin{bmatrix} 4 & 3 & 4 \ -7 & -1 & -5 \end{bmatrix}

Dimension Requirement

If matrices do not have the same dimensions, addition or subtraction cannot be performed.

  • Statement:
    No can do if dimensions differ.

Calculator Use

Calculators can perform matrix addition and subtraction efficiently, handling all element-wise operations simultaneously with ease.