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Section 17.1 Addition and Scalar Multiplication

Definition 17.1.1. Matrix addition and scalar multiplication.

Let \(A\) and \(B\) be \(m \times n\) matrices, i.e., \(A=\left(a_{ij}\right)_{m \times n}\) and \(B = \left(b_{ij}\right)_{m \times n}\text{.}\) Then

  1. Matrix addition is defined by \(A+B = \left(a_{ij} + b_{ij}\right)_{m \times n}\text{.}\)

  2. Scalar multiplication is defined by \(kA = \left(k \cdot a_{ij}\right)_{m \times n}\) where \(k\) is a constant.

Thus, to add two matrices of the same size, add up the corresponding entries in each matrix. Matrix addition of matrices not of the same size is not defined. To multiply a matrix by a scalar, multiply each entry in the matrix by that scalar. Note that we now can also subtract two matrices by defining

\begin{equation*} A-B = A+ (-1)B. \end{equation*}

If \(A = \begin{pmatrix} 1 \amp 2 \amp 3\\ 4 \amp 3 \amp 2\end{pmatrix}, \quad B = \begin{pmatrix} 5 \amp 3 \\ 9 \amp 2 \\ 4 \amp 3\end{pmatrix}.\) Then calculate the given expression or explain why it does not exist.

  1. \(\displaystyle 3A+B\)

  2. \((2A^T - B)_{32} \, \) (i.e. the entry in row \(3\text{,}\) column \(2\) of the matrix \(2A^T-B\,\))

  3. \(\displaystyle (B^T+A)_{31}\)

Solution.

  1. Since \(A\) is a \(2 \times 3\) matrix so will be \(3A\text{.}\) Since \(3A\) and \(B\) are not the same size we cannot add these two matrices.

  2. \(A^T = \begin{pmatrix} 1 \amp 4 \\ 2 \amp 3 \\ 3\amp 2\end{pmatrix}\) and so \(2A^T = \begin{pmatrix} 2 \amp 8 \\ 4 \amp 6 \\ 6 \amp 4\end{pmatrix}\text{.}\) Thus \(2A^T-B = \begin{pmatrix} -3 \amp 5 \\ -5 \amp 4 \\ 2 \amp 1\end{pmatrix}\) and \((2A^T - B)_{32} =1.\)

  3. \(B^T + A = \begin{pmatrix} 5 \amp 9 \amp 4 \\ 3 \amp 2 \amp 3\end{pmatrix} + \begin{pmatrix} 1 \amp 2 \amp 3\\ 4 \amp 3 \amp 2\end{pmatrix} = \begin{pmatrix} 6 \amp 11 \amp 7\\ 7 \amp 5 \amp 5\end{pmatrix}\text{.}\) Since this is a \(2 \times 3\) matrix there is no entry in row \(3\text{,}\) column \(1\text{.}\)

From the definitions of matrix addition and scalar multiplication the following general properties can be shown.

Exercises Example Tasks

1.

Let

\begin{equation*} A = \begin{pmatrix}4 \amp -2 \amp 1\\ 0 \amp 2 \amp 3\end{pmatrix} \quad \mbox{and} \quad B=\begin{pmatrix}1 \amp 2 \\ 3 \amp 4 \\ 5 \amp 6\end{pmatrix}. \end{equation*}

  1. Find \((2A-B^{T})_{12}\text{.}\)

  2. Find \((2A-B)^T\text{.}\)

2.

Show that for any matrix \(A\,\text{,}\) \(3A = A + A + A\text{.}\) For what values of \(k\) is it true that \(kA = \underbrace{A + A + \cdots + A}_{k \mbox{ times}}\,\text{?}\)

3.

Prove that for two matrices \(A\) and \(B\) of the same size \(A+B = B+A\text{.}\)