Mathematics
Grade 11
15 min
Matrix operation rules
Matrix operation rules
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1
Introduction & Learning Objectives
Learning Objectives
Determine if two matrices can be added, subtracted, or multiplied based on their dimensions.
Accurately perform matrix addition and subtraction on compatible matrices.
Perform scalar multiplication on any given matrix.
Correctly perform matrix multiplication using the row-by-column method.
Identify and apply the properties of matrix operations, such as associativity and distributivity.
Solve simple matrix equations involving addition, subtraction, and scalar multiplication.
How do video games transform a 3D character model to move across your 2D screen? It's all done with the 'algebra' of matrices! 🎮
This tutorial introduces the fundamental rules for working with matrices. Just like numbers have rules for addition and multiplication, matr...
2
Key Concepts & Vocabulary
TermDefinitionExample
MatrixA rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is denoted by a capital letter.A = \begin{pmatrix} 2 & -1 \\ 0 & 5 \end{pmatrix} is a 2x2 matrix.
Dimensions (Order)The size of a matrix, described by its number of rows and columns, written as 'rows x columns'.The matrix \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} has dimensions 2x3 because it has 2 rows and 3 columns.
Element (Entry)A single value within a matrix, identified by its row and column position. The element in row 'i' and column 'j' of matrix A is denoted as a_{ij}.In matrix A = \begin{pmatrix} 9 & 8 \\ 7 & 6 \end{pmatrix}, the element a_{21} is 7.
ScalarAn ordinary number (a real number) t...
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Core Formulas
Matrix Addition and Subtraction
If A and B are matrices with the same dimensions (m x n), then their sum A + B and difference A - B are also m x n matrices where each element is the sum or difference of the corresponding elements. \\ [A \pm B]_{ij} = [A]_{ij} \pm [B]_{ij}
To add or subtract matrices, they MUST have the exact same dimensions. You then add or subtract the elements in the same positions.
Scalar Multiplication
If c is a scalar and A is a matrix, the scalar product cA is the matrix obtained by multiplying every element of A by c. \\ [cA]_{ij} = c \cdot [A]_{ij}
To perform scalar multiplication, you simply multiply every single number inside the matrix by the scalar value outside of it.
Matrix Multiplication
If A is an m x p matrix and B is a p x n matrix, t...
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Challenging
If the product of two matrices, AB, results in a 3x3 square matrix, which of the following could be the dimensions of A and B?
A.is 3x2, B is 3x2
B.is 3x5, B is 5x3
C.is 3x3, B is 2x3
D.is 5x3, B is 3x5
Challenging
A student is asked to calculate 2A + B where A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} and B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. The student's work is shown below:
Step 1: 2A = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}
Step 2: 2A + B = \begin{pmatrix} 2*0 & 4*1 \\ 6*1 & 8*0 \end{pmatrix}
Step 3: Result = \begin{pmatrix} 0 & 4 \\ 6 & 0 \end{pmatrix}
Which step contains the first error?
A.Step 1
B.Step 2
C.Step 3
D.There is no error.
Challenging
Given matrices A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} and B = \begin{pmatrix} 4 & 1 \\ 5 & 2 \end{pmatrix}, which of the following statements is true?
A.AB = BA
B.AB is a zero matrix.
C.AB is defined but BA is not.
D.AB ≠BA
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