Answer :
Let's solve the given problems step by step.
### i) Determining the dimensions of matrix [tex]\( P \)[/tex]
The matrix [tex]\( P \)[/tex] is given as:
[tex]\[ P = \left[\begin{array}{ccc} 4 & 1 & 3 \\ -2 & 5 & 6 \\ 7 & 0 & 9 \\ 8 & -7 & -3 \end{array}\right] \][/tex]
From the matrix, we see:
- The number of rows is 4.
- The number of columns is 3.
Thus, the values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] are:
[tex]\[ i = 4 \][/tex]
[tex]\[ j = 3 \][/tex]
### ii) Listing all possible values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex]
The values of [tex]\( i \)[/tex] correspond to the rows, and the values of [tex]\( j \)[/tex] correspond to the columns:
- Possible values of [tex]\( i \)[/tex]: [tex]\( \{1, 2, 3, 4\} \)[/tex]
- Possible values of [tex]\( j \)[/tex]: [tex]\( \{1, 2, 3\} \)[/tex]
So, all possible pairs [tex]\((i, j)\)[/tex] are:
[tex]\[ \{1, 2, 3, 4\} \text{ for } i \][/tex]
[tex]\[ \{1, 2, 3\} \text{ for } j \][/tex]
### iii) Finding [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{13} + p_{ij} = -4 \)[/tex]
From the matrix [tex]\( P \)[/tex], the element [tex]\( p_{13} \)[/tex] is the element in the 1st row and 3rd column, which is:
[tex]\[ p_{13} = 3 \][/tex]
We need to find positions [tex]\( (i, j) \)[/tex] such that:
[tex]\[ 3 + p_{ij} = -4 \][/tex]
[tex]\[ p_{ij} = -4 - 3 \][/tex]
[tex]\[ p_{ij} = -7 \][/tex]
Now, we look for elements in the matrix [tex]\( P \)[/tex] that equal [tex]\(-7\)[/tex]:
In the given matrix:
[tex]\[ P = \left[\begin{array}{ccc} 4 & 1 & 3 \\ -2 & 5 & 6 \\ 7 & 0 & 9 \\ 8 & -7 & -3 \end{array}\right] \][/tex]
The element [tex]\(-7\)[/tex] is located in the 4th row and 2nd column, so:
[tex]\[ (i, j) = (4, 2) \][/tex]
### iv) Finding [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{ij} + p_{43} = -p_{33} \)[/tex]
From the matrix [tex]\( P \)[/tex], we have:
- [tex]\( p_{43} \)[/tex] is the element in the 4th row and 3rd column, which is:
[tex]\[ p_{43} = -3 \][/tex]
- [tex]\( p_{33} \)[/tex] is the element in the 3rd row and 3rd column, which is:
[tex]\[ p_{33} = 9 \][/tex]
We need to find positions [tex]\( (i, j) \)[/tex] such that:
[tex]\[ p_{ij} + (-3) = -9 \][/tex]
[tex]\[ p_{ij} - 3 = -9 \][/tex]
[tex]\[ p_{ij} = -6 \][/tex]
Now, we look for elements in the matrix [tex]\( P \)[/tex] that equal [tex]\(-6\)[/tex]. However, there are no elements in the matrix [tex]\( P \)[/tex] that are equal to [tex]\(-6\)[/tex].
So, there are no values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] that satisfy this condition.
### Summary of Findings:
i) The values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] are [tex]\( 4 \)[/tex] and [tex]\( 3 \)[/tex], respectively.
ii) All possible values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] are:
[tex]\[ i \in \{1, 2, 3, 4\} \][/tex]
[tex]\[ j \in \{1, 2, 3\} \][/tex]
iii) The values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{13} + p_{ij} = -4 \)[/tex] are [tex]\( (4, 2) \)[/tex].
iv) There are no values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{ij} + p_{43} = -p_{33} \)[/tex].
### i) Determining the dimensions of matrix [tex]\( P \)[/tex]
The matrix [tex]\( P \)[/tex] is given as:
[tex]\[ P = \left[\begin{array}{ccc} 4 & 1 & 3 \\ -2 & 5 & 6 \\ 7 & 0 & 9 \\ 8 & -7 & -3 \end{array}\right] \][/tex]
From the matrix, we see:
- The number of rows is 4.
- The number of columns is 3.
Thus, the values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] are:
[tex]\[ i = 4 \][/tex]
[tex]\[ j = 3 \][/tex]
### ii) Listing all possible values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex]
The values of [tex]\( i \)[/tex] correspond to the rows, and the values of [tex]\( j \)[/tex] correspond to the columns:
- Possible values of [tex]\( i \)[/tex]: [tex]\( \{1, 2, 3, 4\} \)[/tex]
- Possible values of [tex]\( j \)[/tex]: [tex]\( \{1, 2, 3\} \)[/tex]
So, all possible pairs [tex]\((i, j)\)[/tex] are:
[tex]\[ \{1, 2, 3, 4\} \text{ for } i \][/tex]
[tex]\[ \{1, 2, 3\} \text{ for } j \][/tex]
### iii) Finding [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{13} + p_{ij} = -4 \)[/tex]
From the matrix [tex]\( P \)[/tex], the element [tex]\( p_{13} \)[/tex] is the element in the 1st row and 3rd column, which is:
[tex]\[ p_{13} = 3 \][/tex]
We need to find positions [tex]\( (i, j) \)[/tex] such that:
[tex]\[ 3 + p_{ij} = -4 \][/tex]
[tex]\[ p_{ij} = -4 - 3 \][/tex]
[tex]\[ p_{ij} = -7 \][/tex]
Now, we look for elements in the matrix [tex]\( P \)[/tex] that equal [tex]\(-7\)[/tex]:
In the given matrix:
[tex]\[ P = \left[\begin{array}{ccc} 4 & 1 & 3 \\ -2 & 5 & 6 \\ 7 & 0 & 9 \\ 8 & -7 & -3 \end{array}\right] \][/tex]
The element [tex]\(-7\)[/tex] is located in the 4th row and 2nd column, so:
[tex]\[ (i, j) = (4, 2) \][/tex]
### iv) Finding [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{ij} + p_{43} = -p_{33} \)[/tex]
From the matrix [tex]\( P \)[/tex], we have:
- [tex]\( p_{43} \)[/tex] is the element in the 4th row and 3rd column, which is:
[tex]\[ p_{43} = -3 \][/tex]
- [tex]\( p_{33} \)[/tex] is the element in the 3rd row and 3rd column, which is:
[tex]\[ p_{33} = 9 \][/tex]
We need to find positions [tex]\( (i, j) \)[/tex] such that:
[tex]\[ p_{ij} + (-3) = -9 \][/tex]
[tex]\[ p_{ij} - 3 = -9 \][/tex]
[tex]\[ p_{ij} = -6 \][/tex]
Now, we look for elements in the matrix [tex]\( P \)[/tex] that equal [tex]\(-6\)[/tex]. However, there are no elements in the matrix [tex]\( P \)[/tex] that are equal to [tex]\(-6\)[/tex].
So, there are no values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] that satisfy this condition.
### Summary of Findings:
i) The values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] are [tex]\( 4 \)[/tex] and [tex]\( 3 \)[/tex], respectively.
ii) All possible values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] are:
[tex]\[ i \in \{1, 2, 3, 4\} \][/tex]
[tex]\[ j \in \{1, 2, 3\} \][/tex]
iii) The values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{13} + p_{ij} = -4 \)[/tex] are [tex]\( (4, 2) \)[/tex].
iv) There are no values of [tex]\( i \)[/tex] and [tex]\( j \)[/tex] such that [tex]\( p_{ij} + p_{43} = -p_{33} \)[/tex].
