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Properties of matrix multiplication
- Intro Lesson: a6:25
- Intro Lesson: b8:14
- Intro Lesson: c5:25
- Lesson: 1a16:29
- Lesson: 1b10:46
- Lesson: 1c14:32
- Lesson: 1d3:21
- Lesson: 1e4:09
- Lesson: 2a6:46
- Lesson: 2b9:31
- Lesson: 3a3:45
- Lesson: 3b4:42
- Lesson: 3c2:48
- Lesson: 3d3:10
- Lesson: 3e4:17
- Lesson: 3f3:22
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
Let X,Y,Z be matrices, In be an identity matrix, and On 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
(XY)Z=X(YZ)
Distributive property
X(Y+Z)=XY+XZ
(Y+Z)X=YX+ZX
There are also some matrix to matrix multiplication properties with zero matrices and identity matrices.
Matrix to matrix multiplication property for the zero matrix
OX=O or XO=O
Matrix to matrix multiplication property for the identity matrix
XIn=X or InX=X
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, XY≠YX.
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.
Associative property
(XY)Z=X(YZ)
Distributive property
X(Y+Z)=XY+XZ
(Y+Z)X=YX+ZX
There are also some matrix to matrix multiplication properties with zero matrices and identity matrices.
Matrix to matrix multiplication property for the zero matrix
OX=O or XO=O
Matrix to matrix multiplication property for the identity matrix
XIn=X or InX=X
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, XY≠YX.
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.
- IntroductionProperties of matrix to matrix multiplication overview:a)The basic matrix to matrix multiplication propertiesb)Failure of the Commutative propertyc)Dimension property
- 1.Verifying the properties of matrix to matrix multiplication
You are given that. Verify that:
a)(XY)Z=X(YZ)b)X(Y+Z)=XY+XZc)(Y+Z)X=YX+ZXd)OX=Oe)IY=Y - 2.Showing that the Commutative property fails
You are given that. Show that:
a)XY≠YXb)X(Y+Z)≠(Y+Z)X - 3.Dimension Property
You are given that X is a 2 x 4 matrix, Y is a 3 x 3 matrix, and Z is a 4 x 3 matrix. Are the following defined? If it is defined, show the dimensions of the matrix.a)XYb)XZc)ZXd)ZYe)XZYf)ZYX