Mathematics
Grade 11
15 min
Multiply two matrices
Multiply two matrices
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1
Introduction & Learning Objectives
Learning Objectives
Determine if two matrices can be multiplied by checking their dimensions.
State the dimensions of the resulting product matrix before performing any calculations.
Accurately calculate the product of two matrices using the row-by-column method.
Compute the dot product of a row vector and a column vector as a fundamental step in matrix multiplication.
Recognize and demonstrate that matrix multiplication is not commutative (i.e., AB ≠ BA in most cases).
Apply matrix multiplication to solve simple problems involving transformations or systems.
Ever wondered how a computer graphics program can rotate, scale, and transform a 3D model with a single click? 💻 The secret lies in the power of multiplying matrices!
This tutorial will guide you through the process o...
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Key Concepts & Vocabulary
TermDefinitionExample
Matrix Dimensions (Order)The size of a matrix, expressed as 'rows x columns'. An m x n matrix has m rows and n columns.The matrix A = [[1, 2, 3], [4, 5, 6]] has dimensions 2 x 3 because it has 2 rows and 3 columns.
Compatibility for MultiplicationThe condition that must be met to multiply two matrices. The number of columns in the first matrix must be equal to the number of rows in the second matrix.A 2x3 matrix can be multiplied by a 3x4 matrix because the inner dimensions (3 and 3) match. However, a 3x4 matrix cannot be multiplied by a 2x3 matrix.
Product MatrixThe resulting matrix obtained after multiplying two matrices. Its dimensions are determined by the outer dimensions of the original matrices.If you multiply a 2x3 matrix by a 3x4 matrix, the produc...
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Core Formulas
The Compatibility Rule
To find the product AB, if matrix A has dimensions m x n, matrix B must have dimensions n x p.
This is the first and most important check. The number of columns of the first matrix (n) must exactly match the number of rows of the second matrix (n). If they don't match, the multiplication is undefined.
Dimensions of the Product Matrix
If A is an m x n matrix and B is an n x p matrix, the product matrix C = AB will be an m x p matrix.
The resulting matrix takes its number of rows from the first matrix (m) and its number of columns from the second matrix (p). The 'inner' dimensions (n) disappear.
The Entry Calculation Formula
The entry in the i-th row and j-th column of the product matrix C = AB, denoted C_{ij}, is the dot product of...
4 more steps in this tutorial
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Challenging
Given A = [[1, -1], [1, -1]], which of the following non-zero matrices B results in the product AB being the 2x2 zero matrix?
A.[[1, 1], [-1, -1]]
B.[[1, 0], [0, 1]]
C.[[2, 3], [2, 3]]
D.[[-1, 1], [-1, 1]]
Challenging
Given A = [[1, 1], [0, 0]], B = [[1, 0], [0, 2]], and C = [[0, 1], [1, 1]], it is found that AB = AC. What is the most accurate conclusion from this specific example?
A.For matrix multiplication, if AB = AC, then B must equal C.
B.Matrix A must be the identity matrix.
C.The general rule 'if AB = AC, then B = C' is false for matrices.
D.Matrix multiplication is commutative.
Challenging
Let A = [[0, 1], [-1, 0]]. Find the matrix A³.
A.[[1, 0], [0, 1]]
B.[[-1, 0], [0, -1]]
C.[[0, 1], [-1, 0]]
D.[[0, -1], [1, 0]]
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