Multiplication of matrices properties

In summary, the conversation discusses the use of the unit column ej and its application in matrix multiplication. The equations provided show that the rows and columns of the product AB are combinations of the rows and columns of A and B, respectively. The unit column ej is represented as (δij)T, with a value of 1 in the jth position and 0 elsewhere. The solution for Aej is a linear combination of a column of A and the entries of ej as scalars. For part c, a formula can be deduced for B being a column vector instead of a matrix.
  • #1
bonfire09
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Homework Statement


Let ej denote the jth unit column that contains a 1 in the jth
position and zeros everywhere else. For a general matrix An×n, describe the following products. (a) Aej (c) eTiAej?

Homework Equations


Rows and Columns of a Product
Suppose that A = [aij] is m × p and B = [bij] is p × n.
• [AB]i∗ = Ai∗B [( ith row of AB)=( ith row of A) ×B]. (3.5.4)

• [AB]∗j = AB∗j [ (jth col of AB)=A× ( jth col of B)]. (3.5.5)

• [AB]i∗ = ai1B1∗ + ai2B2∗ + · · · +aipBp∗aikBk∗. (3.5.6)

• [AB]∗j = A∗1b1j + A∗2b2j + · · · + A∗pbpjA∗kbkj (3.5.7)

These last two equations show that rows of AB are combinations of
rows of B, while columns of AB are combinations of columns of A.

The Attempt at a Solution


For parts a and c I am not even sure what they are even asking for. When its saying ej is a unit column does that mean like this (1 0 0...0) as an example? For part A wouldn't the solution of Aej just just be a linear combination a column of A and the entries of ej as scalars?
 
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  • #2
Yes, it means that ej = (δij)T, i.e. 1 where i=j and zero elsewhere.
From your matrix multiplication formulae, can you deduce a formula for B being a column vector instead of a matrix?
 

What is the definition of multiplication of matrices?

Multiplication of matrices is a mathematical operation where two matrices are multiplied together to produce a new matrix. It is defined as the process of multiplying the corresponding elements of the rows and columns of the two matrices and summing the products.

What are the properties of multiplication of matrices?

The properties of multiplication of matrices are:

  • Associative property: (AB)C = A(BC)
  • Distributive property: A(B+C) = AB + AC
  • Identity property: AI = IA = A
  • Zero property: A0 = 0A = 0
  • Commutative property: AB = BA (only if both matrices are square)

What is the importance of the identity matrix in multiplication of matrices?

The identity matrix is important in multiplication of matrices because it acts as the neutral element, similar to the number 1 in regular multiplication. When a matrix is multiplied by the identity matrix, it remains unchanged. This property is useful in solving systems of linear equations and in finding the inverse of a matrix.

Can the order of matrices be changed when multiplying them?

No, the order of matrices cannot be changed when multiplying them. The number of columns in the first matrix must be equal to the number of rows in the second matrix in order for the multiplication to be possible. Therefore, if the order is changed, the dimensions of the matrices will not match and the multiplication cannot be performed.

What is the difference between matrix multiplication and scalar multiplication?

The main difference between matrix multiplication and scalar multiplication is that scalar multiplication involves multiplying a matrix by a single number (scalar), while matrix multiplication involves multiplying two matrices together. In scalar multiplication, each element in the matrix is multiplied by the scalar, while in matrix multiplication, the corresponding elements in the rows and columns are multiplied and then summed to produce a new matrix.

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