Properties of matrix multiplication - Matrices
Properties of matrix multiplication
In this section, we will learn about the properties of matrix to matrix multiplication. These properties include the associative property, distributive property, zero and identity matrix property, and the dimension property. You will notice that the commutative property fails for matrix to matrix multiplication. Lastly, you will also learn that multiplying a matrix with another matrix is not always defined. The product of the two matrices is only defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.
Lessons
Notes:
Let be matrices, be an identity matrix, and be a zero matrix. If all five of these matrices have equal dimensions, then we will have the following matrix to matrix multiplication properties:
Associative property
Distributive property
There are also some matrix to matrix multiplication properties with zero matrices and identity matrices.
Matrix to matrix multiplication property for the zero matrix
or
Matrix to matrix multiplication property for the identity matrix
or
Here are some important things to know.
Commutative property fails: Notice that the commutative property fails when you use matrix to matrix multiplication. For example, .
Dimension property: When multiplying a matrix with another matrix, it is not always defined. The product of the two matrices is only defined if the number of columns in the first matrix is equal to the number of rows of the second matrix.
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Intro Lesson
Properties of matrix to matrix multiplication overview:
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1.
Verifying the properties of matrix to matrix multiplication
You are given that. Verify that:
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2.
Showing that the Commutative property fails
You are given that. Show that:
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3.
Dimension Property
You are given that is a 2 x 4 matrix, is a 3 x 3 matrix, and is a 4 x 3 matrix. Are the following defined? If it is defined, show the dimensions of the matrix.
