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  5. Modelling With Matrices

Modelling With Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Each number inside the matrix is called an entry. Matrices are often enclosed in square brackets.

The dimension of a matrix is given by multiplying the number of rows by the number of columns. This can be expressed algebraically as (\(m × n\)).

Categorizing Matrices

  • Row Matrix: A matrix with only one row

    Example: \(\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\)


  • Column Matrix: A matrix with only one column

    Example: \(\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)


  • Square Matrix: A matrix with the same number of rows and columns (\(n × n\))

    Example: \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)


  • Identity Matrix: A square matrix with \(1\)s on the diagonal and \(0\)s elsewhere

    Example: \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)


  • Zero Matrix: A matrix where all entries are \(0\)

    Example: \( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

Transpose of a Matrix

  • The transpose of a matrix is obtained by swapping its rows and columns.
  • It is denoted as AT.
  • Steps to find the transpose:
    1. Take the first row and make it the first column
    2. Take the second row and make it the second column
    3. Repeat this for all rows

Matrix Addition and Subtraction

  • Two matrices can be added or subtracted only if they have the same dimensions
  • Addition and subtraction are done element-wise

Steps for Matrix Addition

  1. Ensure both matrices have the same dimensions
  2. Add corresponding elements
  3. Write the result as a new matrix
\[ A + B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \]

Steps for Matrix Subtraction

  1. Ensure both matrices have the same dimensions
  2. Subtract corresponding elements
  3. Write the result as a new matrix
\[A - B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} - \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \\ a_{21} - b_{21} & a_{22} - b_{22} \end{bmatrix}\]


Add the following sets of matrices:

\(A = \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix} \)

We can confirm these matrices can be added since they both have dimensions of \(2\times2\).

Next, we can add these matrices by adding the corresponding elements of each matrix:

\(A + B = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix}\)

\(A + B = \begin{bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \end{bmatrix}\)

\(A + B = \begin{bmatrix}6 & 8 \\ 10 & 12 \end{bmatrix}\)

Therefore, we can determine the sum of matrices \(A\) and \(B\) is \(\begin{bmatrix}6 & 8 \\ 10 & 12 \end{bmatrix}\).

\(C = \begin{bmatrix} -3 & 7 \\ 2 & -5\end{bmatrix}, \quad D = \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix} \)

We can confirm these matrices can be added since they both have dimensions of \(2 \times 2\).

Next, we can add these matrices by adding the corresponding elements of each matrix:

\(C + D = \begin{bmatrix}c_{11} & c_{12} \\ c_{21} & c_{22}\end{bmatrix} + \begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{bmatrix} = \begin{bmatrix} c_{11} + d_{11} & c_{12} + d_{12} \\ c_{21} + d_{21} & c_{22} + d_{22}\end{bmatrix}\)

\(C + D = \begin{bmatrix} -3 + 4 & 7 + (-2) \\ 2 + (-1) & -5 + 3\end{bmatrix}\)

\(C + D = \begin{bmatrix} 1 & 5 \\ 1 & -2 \end{bmatrix}\)

Therefore, we can determine the sum of matrices \(C\) and \(C\) is \(\begin{bmatrix} 1 & 5 \\ 1 & -2 \end{bmatrix}\).

\(M = \begin{bmatrix} 10 & -15 \\ 6 & 9\end{bmatrix}, \quad N = \begin{bmatrix}-8 & 20 \\ -3 & -7\end{bmatrix} \)

We can confirm these matrices can be added since they both have dimensions of \(2 \times 2\).

Next, we can add these matrices by adding the corresponding elements of each matrix:

\(M + N = \begin{bmatrix}m_{11} & m_{12} \\ m_{21} & m_{22}\end{bmatrix} + \begin{bmatrix} n_{11} & n_{12} \\ n_{21} & n_{22} \end{bmatrix} = \begin{bmatrix} m_{11} + n_{11} & m_{12} + n_{12} \\ m_{21} + n_{21} & m_{22} + n_{22}\end{bmatrix}\)

\(M + N = \begin{bmatrix} 10 + (-8) & -15 + 20 \\ 6 + (-3) & 9 + (-7)\end{bmatrix}\)

\(M + N = \begin{bmatrix} 2 & 5 \\ 3 & 2 \end{bmatrix}\)

Therefore, we can determine the sum of matrices \(M\) and \(N\) is \(\begin{bmatrix} 2 & 5 \\ 3 & 2 \end{bmatrix}\).

Subtract the following sets of matrices:

\(E = \begin{bmatrix} 8 & 6 \\ 4 & 2\end{bmatrix}, \quad F = \begin{bmatrix}3 & 2 \\ 1 & 1\end{bmatrix}, \quad E - F\)

We can confirm these matrices can be added since they both have dimensions of \(2 \times 2\).

Next, we can subtract these matrices by subtracting the corresponding elements of each matrix:

\(E - F = \begin{bmatrix} e_{11} & e_{12} \\ e_{21} & e_{22} \end{bmatrix} - \begin{bmatrix} f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix} = \begin{bmatrix} e_{11} - f_{11} & e_{12} - f_{12} \\ e_{21} - f_{21} & e_{22} - f_{22} \end{bmatrix}\)

\(E - F = \begin{bmatrix} 8 - 3 & 6 - 2 \\ 4 - 1 & 2 - 1 \end{bmatrix}\)

\(E - F = \begin{bmatrix} 5 & 4 \\ 3 & 1\end{bmatrix}\)

Therefore, we can determine the difference between matrices \(E\) and \(F\) is \(\begin{bmatrix} 5 & 4 \\ 3 & 1\end{bmatrix}\).

\(X = \begin{bmatrix}-2 & 7 \\ 5 & -3 \end{bmatrix}, \quad Y = \begin{bmatrix}4 & -6 \\ -2 & 8 \end{bmatrix}, \quad X - Y\)

We can confirm these matrices can be added since they both have dimensions of \(2 \times 2\).

Next, we can subtract these matrices by subtracting the corresponding elements of each matrix:

\(X - Y = \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} - \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} = \begin{bmatrix} x_{11} - y_{11} & x_{12} - y_{12} \\ x_{21} - y_{21} & x_{22} - y_{22} \end{bmatrix}\)

\(X - Y = \begin{bmatrix} -2 - 4 & 7 - (-6) \\ 5 - (-2) & -3 - 8 \end{bmatrix}\)

\(X - Y = \begin{bmatrix} -6 & 13 \\ 7 & -11 \end{bmatrix}\)

Therefore, we can determine the difference between matrices \(X\) and \(Y\) is \(\begin{bmatrix} -6 & 13 \\ 7 & -11 \end{bmatrix}\).

\(S = \begin{bmatrix}12 & -5 \\ 9 & 14\end{bmatrix}, \quad T = \begin{bmatrix}7 & 3 \\ -6 & 10 \end{bmatrix}, \quad S - T\)

We can confirm these matrices can be added since they both have dimensions of \(2 \times 2\).

Next, we can subtract these matrices by subtracting the corresponding elements of each matrix:

\(S - T = \begin{bmatrix} s_{11} & s_{12} \\ s_{21} & s_{22} \end{bmatrix} - \begin{bmatrix} t_{11} & t_{12} \\ t_{21} & t_{22} \end{bmatrix} = \begin{bmatrix} s_{11} - t_{11} & s_{12} - t_{12} \\ s_{21} - t_{21} & s_{22} - t_{22} \end{bmatrix}\)

\(S - T = \begin{bmatrix} 12 - 7 & -5 - 3 \\ 9 - (-6) & 14 - 10 \end{bmatrix}\)

\(S - T = \begin{bmatrix}5 & -8 \\ 15 & 4\end{bmatrix}\)

Therefore, we can determine the difference between matrices \(S\) and \(T\) is \(\begin{bmatrix}5 & -8 \\15 & 4\end{bmatrix}\).