Matrices and Operations on Them¶

Matrix¶

Any rectangular arrangement of real numbers consisting of a number of rows and columns is called a matrix. Each real number located in a matrix is called an entry of that matrix.

\(\begin{equation*} A = \begin{bmatrix} 12 & 78 & -120 \\ 3.7 & -809 & 5 \end{bmatrix} \end{equation*}\) \(\begin{equation*} B = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix} \end{equation*}\) \(\begin{equation*} C = \begin{bmatrix} -0.5 & -90 \\ 0 & 4.09 \\ 5 & 6 \\ 92 & 37 \end{bmatrix} \end{equation*}\)

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Order of the Matrix¶

A matrix with m columns and n rows is called an order of n × m.

Matrix Addition and Subtraction¶

To add two matrices, first we must ensure they have the same number of rows and columns (the number of rows doesn't need to equal the number of columns). Then, each entry in matrix A is added to its corresponding entry in matrix B to create the corresponding entry in matrix C. The subtraction operation follows the same method.

\(\begin{equation*} \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} + \begin{bmatrix} 3 & 2 \\ 4 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 5 & 4 \end{bmatrix} \end{equation*}\)

\(\begin{equation*} \begin{bmatrix} 37 & 7 \\ 0 & 3 \end{bmatrix} - \begin{bmatrix} 6 & 13 \\ 2 & -5 \end{bmatrix} = \begin{bmatrix} 31 & -6 \\ -2 & 8 \end{bmatrix} \end{equation*}\)

Matrix Multiplication¶

To multiply matrices, first note that if matrix A has n rows and m columns, matrix B must have m rows and z columns. In this case, the resulting matrix C—the product of the two matrices—will have n rows and z columns. The entry \(c_{ij}\) equals the sum of \(a_{ik} × b_{kj}\), where k is a natural number with a maximum value of m.

\(\begin{equation*} \begin{bmatrix} 4 & 2 & 3 \\ 0 & 3 & -1 \end{bmatrix} \begin{bmatrix} 7 \\ -5 \\ 10 \end{bmatrix} = \begin{bmatrix} 48 \\ -25 \end{bmatrix} \end{equation*}\)

Multiplication of a Number by a Matrix¶

When a real number k is multiplied by a matrix, all entries of the matrix are multiplied by k.

\(\begin{equation*} 5 \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & 5 \\ 10 & 15 \end{bmatrix} \end{equation*}\)