Mathematics
Grade 12
15 min
Multiply a matrix by a scalar
Multiply a matrix by a scalar
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Introduction & Learning Objectives
Learning Objectives
Define scalar multiplication and distinguish it from matrix multiplication.
Perform scalar multiplication on matrices of any dimension with integer, fractional, and negative scalars.
Identify and apply the distributive and associative properties of scalar multiplication to simplify matrix expressions.
Solve linear matrix equations that involve scalar multiplication, matrix addition, and matrix subtraction.
Interpret the geometric effect of scalar multiplication on a vector represented as a column matrix.
Construct and evaluate linear combinations of matrices using scalar multiplication.
Ever used a graphics editor to resize an image by 200% or make it 50% smaller? 🖼️ You've intuitively used the core concept of scalar multiplication!
This tutorial wi...
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Key Concepts & Vocabulary
TermDefinitionExample
ScalarA single real number that is used to multiply a matrix. It is not a matrix itself.The numbers 7, -3, 0.5, and π are all scalars.
MatrixA rectangular array or grid of numbers, symbols, or expressions, arranged in rows and columns.A = [[2, -1], [0, 5]] is a 2x2 matrix.
Element (or Entry)An individual value located at a specific row and column within a matrix.In the matrix A = [[2, -1], [0, 5]], the element in the first row, second column (a₁₂) is -1.
Scalar MultiplicationThe operation of multiplying a matrix by a scalar, where every element in the matrix is multiplied by that scalar.If c = 3 and A = [[1, 2], [3, 4]], then cA = [[3*1, 3*2], [3*3, 3*4]] = [[3, 6], [9, 12]].
Resultant MatrixThe matrix that is the outcome of a matrix operation. In scalar multiplicati...
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Core Formulas
The Rule of Scalar Multiplication
If c is a scalar and A is an m x n matrix with elements a_ij, then the product cA is an m x n matrix whose elements are c * a_ij.
This is the fundamental definition. To multiply a matrix by a scalar, you simply multiply every single element inside the matrix by that scalar. The dimensions of the matrix do not change.
Distributive Property (Scalar over Matrix Addition)
c(A + B) = cA + cB
You can either add the matrices A and B first and then multiply the result by the scalar c, or you can multiply each matrix by the scalar c first and then add the resulting matrices. The outcome is the same.
Associative Property of Scalar Multiplication
(cd)A = c(dA)
If you are multiplying a matrix by two scalars (c and d), you can either multiply the...
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Challenging
Solve for the 2x1 matrix X in the following equation: 4(X + [[-1], [2]]) = 2(X + [[3], [0]])
A.[[5], [-4]]
B.[[1], [-2]]
C.[[2], [-1]]
D.[[4], [-3]]
Challenging
If X, A, and B are matrices of the same dimension, and c and d are non-zero scalars, solve the equation cX + A = dX + B for X in terms of A, B, c, and d.
A.X = (c-d)(B-A)
B.X = (c-d)⁻¹(A-B)
C.X = (c-d)⁻¹(B-A)
D.X = (d-c)⁻¹(B-A)
Challenging
Given the system of matrix equations: X + Y = [[5, 2], [1, 6]] and 2X - Y = [[4, 7], [5, 3]]. Solve for matrix X.
A.[[9, 9], [6, 9]]
B.[[3, 3], [2, 3]]
C.[[2, -1], [-1, 3]]
D.[[1, -5], [-4, 3]]
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