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Linear Algebra For Undergraduate Students

Yilan(Peter) Tan

Section 2.2 Matrix Addition and Scalar Multiplication

Just as we can add numbers and multiply them by constants, matrices support similar fundamental operations. These operations—matrix addition and scalar multiplication—form the foundation of matrix algebra and are essential for understanding how matrices behave as mathematical objects.
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In many real-world applications, these operations have natural interpretations. For instance, adding matrices might represent combining datasets, summing forces in physics, or merging financial portfolios. Scalar multiplication could represent scaling a transformation, adjusting all values by a common factor, or changing units of measurement.
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Before diving into the formal definitions, let’s consider a motivating example: Suppose a company operates in three cities and tracks monthly sales data for four product categories. The data can be represented as matrices:
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Example 2.2.1. Business Application: Combining Sales Data.

Quarter 1 sales (in thousands of dollars):
\begin{equation*} Q_1 = \begin{pmatrix} 12 \amp 15 \amp 8 \amp 20 \\ 18 \amp 22 \amp 12 \amp 25 \\ 10 \amp 13 \amp 6 \amp 15 \end{pmatrix} \end{equation*}
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Quarter 2 sales (in thousands of dollars):
\begin{equation*} Q_2 = \begin{pmatrix} 14 \amp 17 \amp 10 \amp 23 \\ 16 \amp 24 \amp 14 \amp 28 \\ 12 \amp 15 \amp 8 \amp 18 \end{pmatrix} \end{equation*}
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To find the total sales for the first half of the year, we naturally want to add corresponding entries: \(Q_1 + Q_2\text{.}\) If the company wants to project next year’s sales assuming a 15% increase across all categories and cities, we would compute \(1.15(Q_1 + Q_2)\text{.}\)
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This example illustrates why matrix addition and scalar multiplication are not just abstract mathematical concepts, but practical tools for organizing and manipulating structured data.
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Subsection 2.2.1 Matrix Addition

Definition 2.2.2. Matrix Addition.

Let \(A = [a_{ij}]\) and \(B = [b_{ij}]\) be two matrices of the same size \(m \times n\text{.}\) The sum of \(A\) and \(B\text{,}\) denoted \(A + B\text{,}\) is the \(m \times n\) matrix whose entries are given by:
\begin{equation*} (A + B)_{ij} = a_{ij} + b_{ij} \end{equation*}
for all \(1 \leq i \leq m\) and \(1 \leq j \leq n\text{.}\)
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Remark 2.2.3.

Important: Matrix addition is only defined for matrices of the same size. You cannot add a \(2 \times 3\) matrix to a \(3 \times 2\) matrix, even though they have the same number of total entries.
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Example 2.2.4. Adding 2×2 Matrices.

Let \(A = \begin{pmatrix} 3 \amp -1 \\ 2 \amp 4 \end{pmatrix}\) and \(B = \begin{pmatrix} -2 \amp 5 \\ 1 \amp -3 \end{pmatrix}\text{.}\)
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Then:
\begin{align*} A + B \amp= \begin{pmatrix} 3 \amp -1 \\ 2 \amp 4 \end{pmatrix} + \begin{pmatrix} -2 \amp 5 \\ 1 \amp -3 \end{pmatrix}\\ \amp= \begin{pmatrix} 3+(-2) \amp (-1)+5 \\ 2+1 \amp 4+(-3) \end{pmatrix}\\ \amp= \begin{pmatrix} 1 \amp 4 \\ 3 \amp 1 \end{pmatrix} \end{align*}
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Example 2.2.5. Adding Larger Matrices.

Consider the matrices:
\begin{equation*} A = \begin{pmatrix} 0 \amp 2 \amp -8 \amp 8 \\ 1 \amp -2 \amp 1 \amp 0 \\ 5 \amp 0 \amp -5 \amp 10 \end{pmatrix}, \quad B = \begin{pmatrix} 0 \amp 3 \amp -6 \amp 6 \\ 3 \amp -7 \amp 8 \amp -5 \\ 3 \amp -9 \amp 12 \amp -9 \end{pmatrix} \end{equation*}
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Computing step by step:
\begin{equation*} A + B = \begin{pmatrix} 0+0 \amp 2+3 \amp (-8)+(-6) \amp 8+6 \\ 1+3 \amp (-2)+(-7) \amp 1+8 \amp 0+(-5) \\ 5+3 \amp 0+(-9) \amp (-5)+12 \amp 10+(-9) \end{pmatrix} = \begin{pmatrix} 0 \amp 5 \amp -14 \amp 14 \\ 4 \amp -9 \amp 9 \amp -5 \\ 8 \amp -9 \amp 7 \amp 1 \end{pmatrix} \end{equation*}
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Example 2.2.6. Completing the Business Example.

Returning to our business example, let’s compute the total sales for the first half of the year:
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The resulting matrix shows total sales by city (rows) and product category (columns) for the first half of the year. For instance, City 1 had total sales of 26 thousand dollars in Product Category 1.
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Insight 2.2.7. Geometric Interpretation.

In \(\mathbb{R}^2\) or \(\mathbb{R}^3\text{,}\) matrix addition corresponds to vector addition. For example, if we interpret \(2 \times 1\) matrices as position vectors in the plane, then adding them corresponds to the familiar parallelogram law of vector addition.
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Subsection 2.2.2 Scalar Multiplication

Definition 2.2.8. Scalar Multiplication.

Let \(A = [a_{ij}]\) be an \(m \times n\) matrix and let \(c\) be a scalar (real number). The scalar multiple of \(A\) by \(c\text{,}\) denoted \(cA\text{,}\) is the \(m \times n\) matrix whose entries are given by:
\begin{equation*} (cA)_{ij} = c \cdot a_{ij} \end{equation*}
for all \(1 \leq i \leq m\) and \(1 \leq j \leq n\text{.}\)
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Example 2.2.9. Basic Scalar Multiplication.

Let \(A = \begin{pmatrix} 2 \amp -3 \\ 1 \amp 4 \\ 0 \amp -2 \end{pmatrix}\) and \(c = -3\text{.}\)
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Then:
\begin{align*} cA = (-3)A \amp= (-3) \begin{pmatrix} 2 \amp -3 \\ 1 \amp 4 \\ 0 \amp -2 \end{pmatrix}\\ \amp= \begin{pmatrix} (-3)(2) \amp (-3)(-3) \\ (-3)(1) \amp (-3)(4) \\ (-3)(0) \amp (-3)(-2) \end{pmatrix}\\ \amp= \begin{pmatrix} -6 \amp 9 \\ -3 \amp -12 \\ 0 \amp 6 \end{pmatrix} \end{align*}
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Example 2.2.10. Computational Example.

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Example 2.2.11. Business Projection with Scalar Multiplication.

Using our business example, suppose the company wants to project sales for the second half of the year, assuming a 15% increase over the first half. This would be computed as \(1.15 \times (Q_1 + Q_2)\text{:}\)
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Insight 2.2.12. Geometric Interpretation.

For vectors in \(\mathbb{R}^2\) or \(\mathbb{R}^3\text{,}\) scalar multiplication has a clear geometric meaning:
  • If \(c > 1\text{,}\) the vector is stretched by factor \(c\)
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  • If \(0 < c < 1\text{,}\) the vector is compressed by factor \(c\)
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  • If \(c < 0\text{,}\) the vector is both scaled by \(|c|\) and reversed in direction
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  • If \(c = 0\text{,}\) the result is the zero vector
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Subsection 2.2.3 Algebraic Properties

Matrix addition and scalar multiplication satisfy many of the same algebraic properties as addition and multiplication of real numbers. These properties are fundamental to working effectively with matrices.
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Example 2.2.15. Verifying Properties.

Let’s verify the distributive property \(c(A + B) = cA + cB\) with specific matrices:
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Subsection 2.2.4 Real-World Applications

Example 2.2.16. Physics: Adding Force Vectors.

In physics, forces are vectors that can be represented as column matrices. Suppose three forces act on an object:
\begin{equation*} \mathbf{F_1} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \quad \mathbf{F_2} = \begin{pmatrix} -2 \\ 1 \end{pmatrix}, \quad \mathbf{F_3} = \begin{pmatrix} 1 \\ -3 \end{pmatrix} \end{equation*}
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The net force is: \(\mathbf{F_{net}} = \mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3} = \begin{pmatrix} 3-2+1 \\ 4+1-3 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}\)
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If we want to find the force needed to achieve twice this net force, we compute \(2\mathbf{F_{net}} = \begin{pmatrix} 4 \\ 4 \end{pmatrix}\text{.}\)
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Example 2.2.17. Economics: Supply and Demand Analysis.

A retailer tracks supply and demand for products across different regions. Monthly supply and demand can be represented as matrices:
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Subsection 2.2.5 Practice Problems

Activity 2.2.1. Matrix Addition Practice.

Given the matrices:
\begin{equation*} A = \begin{pmatrix} 2 \amp -1 \amp 3 \\ 0 \amp 4 \amp -2 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \amp 3 \amp -1 \\ 2 \amp -2 \amp 5 \end{pmatrix}, \quad C = \begin{pmatrix} -1 \amp 2 \\ 3 \amp 0 \end{pmatrix} \end{equation*}
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Compute the following (where possible):
  1. \(\displaystyle A + B\)
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  2. \(\displaystyle B + A\)
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  3. \(\displaystyle A + C\)
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  4. \(\displaystyle (A + B) + (-B)\)
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Activity 2.2.2. Scalar Multiplication Practice.

Using the matrix \(A = \begin{pmatrix} 1 \amp -2 \amp 3 \\ 0 \amp 1 \amp -1 \\ 2 \amp 0 \amp 4 \end{pmatrix}\text{,}\) compute:
  1. \(\displaystyle 3A\)
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  2. \(\displaystyle -2A\)
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  3. \(\displaystyle \frac{1}{2}A\)
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  4. \(\displaystyle 0A\)
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  5. \(2A + 3A\) and \(5A\text{.}\) Are they equal?
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Activity 2.2.3. Combined Operations.

A manufacturing company produces widgets at three factories. The production data for January and February is given by:
\begin{equation*} \text{January} = \begin{pmatrix} 100 \amp 150 \\ 120 \amp 80 \\ 90 \amp 110 \end{pmatrix}, \quad \text{February} = \begin{pmatrix} 110 \amp 140 \\ 130 \amp 85 \\ 95 \amp 120 \end{pmatrix} \end{equation*}
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Each row represents a factory, and each column represents a different widget type.
  1. Find the total production for the two-month period.
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  2. If production increases by 20% in March over February’s levels, what will March production be?
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  3. What is the average monthly production over the three months?
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