(a) Show that if A is an m x n matrix and A(BA) is defined, then B is an n x m matrix.
(b) Show that if A has a row of zeros and B is any matrix for which AB is defined, then AB also has a row of zeros.
(c) Find a similar result involving a column of zeros.
Thanks!!!
(a) Let A be an m x n matrix and B be an n x m matrix. Then A(BA) is defined because the number of columns in A is equal to the number of rows in B.
(b) Let A be an m x n matrix with a row of zeros and B be an n x p matrix. Then AB is defined because the number of columns in A is equal to the number of rows in B. Since A has a row of zeros, the product AB will also have a row of zeros.
(c) Let A be an m x n matrix with a column of zeros and B be an n x p matrix. Then AB is defined because the number of columns in A is equal to the number of rows in B. Since A has a column of zeros, the product AB will also have a column of zeros.
(a) To show that B is an n x m matrix, we need to determine its dimensions based on the given information.
Given that A is an m x n matrix and A(BA) is defined, we can use matrix multiplication rules to determine the dimensions of BA.
Let's break down the multiplication A(BA) step by step:
1. The matrix B must have the same number of columns as A(BA) in order for the multiplication to be defined.
2. The matrix BA must have the same number of rows as A, and the same number of columns as B.
3. Since A has dimensions m x n, B must have dimensions n x p, where p is the number of columns in B.
4. From step 2, we know that BA must have dimensions m x p.
5. Now, we can substitute the dimensions of BA in the expression A(BA):
A(BA) = A(m x p) = (m x n)(n x p) = m x p.
6. Since A(BA) is defined, we can conclude that B must have dimensions p x m.
Therefore, B is an n x m matrix.
(b) To show that AB has a row of zeros when A has a row of zeros, we can prove it by using matrix multiplication rules.
Let A be an m x n matrix with a row of zeros, and let B be any matrix for which AB is defined.
1. Let's denote the row of zeros in matrix A as a vector of zeros, denoted as 0.
2. Assuming matrix B has dimensions n x p, we can write AB as (m x n)(n x p) = (m x p).
3. Let's consider the element at position (i, j) in matrix AB, denoted as (AB)ij.
4. (AB)ij is computed by the dot product of the ith row of A and the jth column of B.
5. Since the ith row of A consists of all zeros, the dot product of this row with any column of B will always be zero.
6. Therefore, for any position (i, j), the element (AB)ij will be zero.
7. Since the element at position (i, j) in the product AB is always zero when A has a row of zeros, we can conclude that AB also has a row of zeros.
(c) To find a similar result involving a column of zeros, we can use similar logic as in part (b) but with transposed matrices.
Let A be an m x n matrix with a column of zeros, and let B be any matrix for which AB is defined.
1. Similar to part (b), we denote the column of zeros in A as a vector of zeros, denoted as 0.
2. Assuming matrix B has dimensions n x p, we can write AB as (m x n)(n x p) = (m x p).
3. Let's consider the element at position (i, j) in matrix AB, denoted as (AB)ij.
4. (AB)ij is computed by the dot product of the ith row of A and the jth column of B.
5. Since A has a column of zeros, this means the jth column of B is multiplied by a vector of zeros, resulting in a zero dot product.
6. Therefore, for any position (i, j), the element (AB)ij will be zero.
7. Since the element at position (i, j) in the product AB is always zero when A has a column of zeros, we can conclude that AB also has a column of zeros.
This is the similar result involving a column of zeros.
(a) To show that if A is an m x n matrix and A(BA) is defined, then B is an n x m matrix, let's start by considering the dimensions of the matrices involved.
Given that A is an m x n matrix, we know that the number of rows in A is m, and the number of columns is n.
Now, considering the product A(BA), the number of columns in BA must be equal to the number of columns in A for the multiplication to be defined.
We know that A is an m x n matrix and BA is defined, so the number of columns in BA must be equal to n.
Since the number of columns in BA is equal to n, we can deduce that the number of columns in B must also be n. Otherwise, the product A(BA) would not be defined.
Therefore, B is an n x m matrix.
(b) To show that if A has a row of zeros and B is any matrix for which AB is defined, then AB also has a row of zeros, let's analyze the multiplication process.
Suppose A is an m x n matrix with a row of zeros. This means that there exists a row vector in A consisting entirely of zeros.
Now, consider the product AB, where B is any matrix for which AB is defined. In matrix multiplication, each element of the resulting matrix AB is obtained by taking the dot product of the corresponding row of A and the corresponding column of B.
Since there is a row of zeros in A, the dot product of that row with any column of B will be zero. This is because the dot product of any vector with a vector of zeros is always zero.
Therefore, every element in the row of AB corresponding to the row of zeros in A will be zero. Hence, AB will have a row of zeros.
(c) To find a similar result involving a column of zeros, let's consider the following:
Suppose A is an m x n matrix with a column of zeros. This means that there exists a column vector in A consisting entirely of zeros.
Now, consider the product AB, where B is any matrix for which AB is defined. In matrix multiplication, each element of the resulting matrix AB is obtained by taking the dot product of the corresponding row of A and the corresponding column of B.
Since there is a column of zeros in A, the dot product of any row of A with that column of zeros will be zero. This is because the dot product of any vector with a vector of zeros is always zero.
Therefore, every element in the column of AB corresponding to the column of zeros in A will be zero. Hence, AB will have a column of zeros.