Mathematics
Grade 11
15 min
Multiply a matrix by a scalar
Multiply a matrix by a scalar
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1
Introduction & Learning Objectives
Learning Objectives
Define a scalar and explain the process of scalar multiplication.
Perform scalar multiplication on matrices of various dimensions (e.g., 2x2, 2x3, 3x2).
Apply the properties of scalar multiplication, including the distributive and associative properties.
Solve simple matrix equations that involve scalar multiplication.
Interpret the result of scalar multiplication in a given real-world context.
Differentiate between scalar multiplication and matrix multiplication.
Ever needed to double a recipe or perfectly resize an image on a computer? Scaling a matrix uses the exact same idea! 📈
In this tutorial, you will learn one of the most fundamental matrix operations: scalar multiplication. This simple process of 'scaling' a matrix is a crucial build...
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Key Concepts & Vocabulary
TermDefinitionExample
ScalarIn the context of matrices, a scalar is simply a single real number (like 5, -1/2, or π) that is used to multiply a matrix.The number 3 in the expression 3A, where A is a matrix.
MatrixA rectangular array of numbers, symbols, or expressions, arranged in rows and columns.A = \begin{pmatrix} 2 & 7 \\ 1 & -4 \end{pmatrix} is a 2x2 matrix.
Element (or Entry)Each individual value within a matrix, identified by its row and column position.In the matrix A = \begin{pmatrix} 2 & 7 \\ 1 & -4 \end{pmatrix}, the element in the first row, second column is 7.
Dimensions of a MatrixThe size of a matrix, described by its number of rows and columns, written as 'rows x columns'.A matrix with 3 rows and 4 columns has dimensions 3x4.
Scalar Multiplication...
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Core Formulas
The Rule of Scalar Multiplication
If k is a scalar and A is a matrix, then the product kA is the matrix found by multiplying each element of A by k. k \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix}
This is the fundamental rule. To perform scalar multiplication, you distribute the scalar to every single element inside the matrix. The dimensions of the matrix do not change.
Distributive Property (Scalar over Matrix Addition)
k(A + B) = kA + kB
For any scalar k and matrices A and B of the same dimensions, you can either add the matrices first and then multiply by the scalar, or multiply each matrix by the scalar first and then add the resulting matrices. The outcome will be identical.
Associative Proper...
4 more steps in this tutorial
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Challenging
Solve for the scalar k in the equation: k \begin{pmatrix} 2 & 1 \ 4 & 0 \end{pmatrix} + \begin{pmatrix} 1 & 3 \ -2 & 5 \end{pmatrix} = \begin{pmatrix} 7 & 6 \ 10 & 5 \end{pmatrix}
A.k = 2
B.k = 4
C.k = 3
D.k = -1
Challenging
If A is a non-zero matrix and k is a scalar such that kA is the zero matrix (a matrix where all elements are 0), what must be true about the scalar k?
A.k must be 1
B.k must be 0
C.k must be -1
D.k can be any real number
Challenging
Let A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} and B = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}. Simplify the expression \frac{1}{2}(4A + 10B) using the distributive property before calculating the final matrix.
A.\begin{pmatrix} 12 & 15 \ 20 & 27 \end{pmatrix}
B.\begin{pmatrix} 2 & 5 \ 10 & 12.5 \end{pmatrix}
C.\begin{pmatrix} 6 & 15 \ 20 & 27 \end{pmatrix}
D.\begin{pmatrix} 12 & 15 \ 20 & 25 \end{pmatrix}
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