Concept explainers
A number of matrices are defined as
Answer the following questions regarding these matrices:
(a) What are the dimensions of the matrices?
(b) Identify the square, column, and row matrices.
(c) What are the values of the elements:
(d) Perform the following operations:
(a)
To calculate: The dimensions of the matrices,
Answer to Problem 2P
Solution:
Dimension of the matrix
Explanation of Solution
Given:
The matrices,
Formula used:
The representation of dimension of any matrix can be written as
Calculation:
Consider the matrix A,
Matrix A has
Therefore, the dimension of the matrix A is
Consider the matrix B,
Matrix B has
Therefore, the dimension of the matrix B is
Consider the matrix C,
Matrix C has
Therefore, the dimension of the matrix C is
Consider the matrix D,
Matrix D has
Therefore, the dimension of the matrix D is
Consider the matrix E,
Matrix E has
Therefore, the dimension of the matrix E is
Consider the matrix F,
The matrix F has
Therefore, the dimension of the matrix F is
Consider the matrix G,
Matrix G has
Therefore, the dimension of the matrix G is
(b)
The square matrix, column matrix and row matrix from the following matrices,
Answer to Problem 2P
Solution:
Matrices B and E are square matrices, matrix C is a column matrix and matrix G is a row matrix.
Explanation of Solution
Given:
The matrices,
Square matrix: A matrix M is said to be square matrix, if number of rows equals to number of columns.
Column Matrix: A matrix M is said to be column matrix, if number of rows can be any natural number but number of columns should be
Row matrix: A matrix M is said to be row matrix, if number of rows should be
Since, the dimension of the matrix B is
So, number of rows equal to number of columns.
Hence,
Since, the dimension of the matrix C is
So, number of rows is
Hence,
Since, the dimension of the matrix E is
So, number of rows equal to number of columns.
Hence,
Since, the dimension of the matrix G is
So, number of rows is
Hence,
Therefore, matrices B and E are square matrices, matrix C is a column matrix and matrix G is a row matrix.
(c)
To calculate: The value of the elements
Answer to Problem 2P
Solution:
The value of the elements
Explanation of Solution
Given:
The matrices,
Formula used:
The value of the element
Calculation:
Consider the matrix A,
Then the value of
Therefore,
Consider the matrix B,
Then the value of
Therefore,
Consider the matrix D,
Then the value of
Since, matrix D does not have
Therefore,
Consider the matrix E,
Then the value of
Therefore,
Consider the matrix F,
Then the value of
Therefore,
Consider the matrix G,
Then the value of
Therefore,
(d)
To calculate: The following operations,
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
Where,
Answer to Problem 2P
Solution:
(1)
(4)
(7)
(10)
Explanation of Solution
Given:
The matrices,
Formula used:
If
Addition of A and B is denoted by
Subtraction of A and B is denoted by
Multiply any scalar quantity
Transpose of the matrix A is denoted by
Multiplication of the matrices A and B is denoted by
Multiplication of two matrices is possible if interior dimensions are equal.
If matrix A has dimension
Calculation:
(1) Consider the matrices E and B,
Now, the addition of two matrices is,
Therefore,
(2) Consider the matrices A and F,
Now, the multiplication of two matrices is,
Therefore,
(3) Consider the matrices B and E,
Now, the subtraction of two matrices is,
Therefore,
(4) Consider a matrix B, and a scalar
The multiplication of a matrix B with a scalar
Therefore,
(5) Consider the matrices E and B,
Now the multiplication of two matrices E and B is,
Therefore,
(6) Consider the matrix C,
Now the transpose of the matrix C is,
Therefore,
(7) Consider the matrices B and A,
Now the multiplication of the two matrices B and A is,
Therefore,
(8) Consider the matrix D,
Now, the transpose of the matrix D is,
Therefore,
(9) Consider the matrices A and C,
Since, interior dimensions of
Therefore,
(10) Consider the identity matrix I of dimension
Since, the multiplication of the identity matrix with any matrix is again that matrix.
Hence,
Therefore,
(11) Consider the matrix E,
Now the transpose of the matrix E is,
The multiplication of the matrix E and its transpose is,
Therefore,
(12) Consider the matrix C,
Now the transpose of the matrix C is,
Multiplication of the matrix C and its transpose is,
Therefore,
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