24.1 Operations On Matrices

1.

Addition:
Two matrices can be added only if they have the same order, and the sum is formed by adding corresponding entries. The matrix \( \begin {pmatrix} 3 & 4 & 5\\ 1 & 0 & 2\\ \end {pmatrix} \) is a \(2\times 3\) matrix, so it can be added only to another \(2\times 3\) matrix.

\[ \begin {pmatrix} 3 & 4 & 5\\ 1 & 2 & 3\\ \end {pmatrix} \pm \begin {pmatrix} 0 & -4 & 6\\ 3 & -2 & 1\\ \end {pmatrix} \]

Let \(A=\\ \begin {pmatrix} 3 & 4 & 6\\ 2 & 5 & 7\\ \end {pmatrix} \) \(\implies \) \(2A= \begin {pmatrix} 6 & 8 & 12\\ 4 & 10 & 14\\ \end {pmatrix} \)

When multiplying a matrix by a constant meaning you multiply each element in the matrix by the constant.

2.

Multiplication Of Matrices
Let \(A= \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} \) and \(B= \begin {pmatrix} e & f\\ g & h\\ \end {pmatrix} \)

Multiplication meaning you multiply the corresponding element of the first matrix (row) to the first column of the second matrix
\[ AB = \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} \begin {pmatrix} e & f\\ g & h\\ \end {pmatrix} = \begin {pmatrix} ae+bg & af+bh\\ ce+dg & cf+dh\\ \end {pmatrix} \]

\[ \text {E.g}\quad \begin {pmatrix} 2 & 3 & 0\\ 1 & 4 & 1\\ \end {pmatrix}_{2\times 3} \begin {pmatrix} 1 & 0 & 2\\ -2 & 3 & 1\\ 1 & 4 & 4\\ \end {pmatrix}_{3\times 3} = \begin {pmatrix} -4 & 9 & 7\\ -6 & 16 & 10\\ \end {pmatrix}_{2\times 3} \]

You can multiply matrices if the column in the first matrix = Row in the second matrix.

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