Addition and Scalar Multiplication

Section 17.1 Addition and Scalar Multiplication

Definition 17.1.1. Matrix addition and scalar multiplication.

Let

A

and

B

m \times n

matrices, i.e.,

A = {(a_{i j})}_{m \times n}

and

B = {(b_{i j})}_{m \times n} .

Then

Matrix addition is defined by $A + B = {(a_{i j} + b_{i j})}_{m \times n} .$
Scalar multiplication is defined by $k A = {(k \cdot a_{i j})}_{m \times n}$ where $k$ is a constant.

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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

A - B = A + (- 1) B .

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Example 17.1.2.

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

$3 A + B$
$(2 A^{T} - B)_{32}$ (i.e. the entry in row $3,$ column $2$ of the matrix $2 A^{T} - B$ )
$(B^{T} + A)_{31}$

Solution.

Since $A$ is a $2 \times 3$ matrix so will be $3 A .$ Since $3 A$ and $B$ are not the same size we cannot add these two matrices.
$A^{T} = (\begin{matrix} 1 & 4 \\ 2 & 3 \\ 3 & 2 \end{matrix})$ and so $2 A^{T} = (\begin{matrix} 2 & 8 \\ 4 & 6 \\ 6 & 4 \end{matrix}) .$ Thus $2 A^{T} - B = (\begin{matrix} - 3 & 5 \\ - 5 & 4 \\ 2 & 1 \end{matrix})$ and $(2 A^{T} - B)_{32} = 1.$
$B^{T} + A = (\begin{matrix} 5 & 9 & 4 \\ 3 & 2 & 3 \end{matrix}) + (\begin{matrix} 1 & 2 & 3 \\ 4 & 3 & 2 \end{matrix}) = (\begin{matrix} 6 & 11 & 7 \\ 7 & 5 & 5 \end{matrix}) .$ Since this is a $2 \times 3$ matrix there is no entry in row $3,$ column $1 .$

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

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Theorem 17.1.3. Properties of Scalar Multiplication and Matrix Addition.

Let $A,$ $B$ and $C$ be matrices of the same size. Then

(A1): $A + B = B + A$ (Commutative Law)
(A2): $(A + B) + C = A + (B + C)$ (Associative Law)
(A3): $A + 0 = A$ (Identity Law)
(A4): $A + (- A) = 0$ (Inverse Law)
(S1): $k (A + B) = k A + k B$
(S2): $(k_{1} + k_{2}) A = k_{1} A + k_{2} A$
(S3): $k_{1} (k_{2} A) = (k_{1} k_{2}) A$
(S4): $1 A = A$

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Exercises Example Tasks

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1.

Let

A = (\begin{matrix} 4 & - 2 & 1 \\ 0 & 2 & 3 \end{matrix}) and B = (\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{matrix}) .

Find $(2 A - B^{T})_{12} .$
Find $(2 A - B)^{T} .$

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2.

Show that for any matrix $A,$ $3 A = A + A + A .$ For what values of $k$ is it true that $k A = \underset{k times}{\underset{⏟}{A + A + \dots + A}} ?$

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3.

Prove that for two matrices $A$ and $B$ of the same size $A + B = B + A .$