Question
Consider the following 3x3 matrix A = [a11 a12 a13; a21 a22 a23; a31 a32 a33] where a11, a12, a13, a...Show More
Solution
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a) To find the determinant of the 3x3 matrix A using the cofactor expansion method, we will expand along the first row (though any row or column can be used). The determinant of A, denoted as det(A), is given by:
$$
\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}
$$
\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}
\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}
$$
where \(C_{ij}\) is the cofactor of element \(a_{ij}\), which is calculated as \(C_{ij} = (-1)^{i+j}M_{ij}\), and \(M_{ij}\) is the minor of \(a_{ij}\), the determinant of the submatrix that remains after removing the \(i\)-th row and \(j\)-th column from A.
The cofactors are:
$$
C_{11} = (-1)^{1+1}\text{det}\begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} = \text{det}\begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} = a_{22}a_{33} - a_{23}a_{32}
$$
$$
C_{12} = (-1)^{1+2}\text{det}\begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} = -\text{det}\begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} = -(a_{21}a_{33} - a_{23}a_{31})
$$
$$
C_{13} = (-1)^{1+3}\text{det}\begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} = \text{det}\begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} = a_{21}a_{32} - a_{22}a_{31}
$$
Substituting these cofactors into the determinant formula, we get:
$$
\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
$$
b) To prove that the determinant of matrix A can also be calculated by multiplying the elements of any row or column by their corresponding cofactors and adding the products, we will use the properties of determinants and cofactors. The determinant of a matrix is a linear function with respect to each row and column. This means that the determinant can be expressed as a sum of products of elements and their corresponding cofactors for any single row or column.
For any element \(a_{ij}\) in row \(i\), the determinant can be expressed as:
$$
\text{det}(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3}
$$
This is true for any row \(i\) and similarly for any column \(j\). The proof of this property is based on the multilinearity and alternating sign properties of the determinant function.
c) Given that the determinant of matrix A is equal to 5, we can use the formula derived in part (a) to find the value of \(a_{11}\) given the other elements:
$$
5 = a_{11}(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7)
$$
Solving for \(a_{11}\):
$$
5 = a_{11}(45 - 48) - 2(36 - 42) + 3(32 - 35)
$$
$$
5 = a_{11}(-3) + 12 - 9
$$
$$
5 = -3a_{11} + 3
$$
$$
2 = -3a_{11}
$$
$$
a_{11} = -\frac{2}{3}
$$
d) To find the inverse of matrix A, denoted as \(A^{-1}\), we use the formula:
$$
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
$$
where adj(A) is the adjugate of A, which is the transpose of the matrix of cofactors of A. Since we have already calculated the cofactors for the first row in part (a), we would need to calculate the cofactors for the remaining elements and then transpose the matrix of cofactors.
However, since the question does not provide all the necessary elements to calculate the full set of cofactors, we cannot complete this step without additional information. If all elements were known, we would calculate the cofactors for each element, form the matrix of cofactors, transpose it to get the adjugate, and then multiply by the reciprocal of the determinant to find the inverse.
e) The cofactor expansion method for finding the determinant and inverse of a 3x3 matrix is a direct method that involves calculating minors and cofactors. It is particularly useful for small matrices where the calculations are manageable. However, for larger matrices, this method becomes cumbersome due to the rapidly increasing number of minors that must be calculated.
In contrast, Gaussian elimination is an algorithmic approach that transforms the matrix into an upper triangular form, from which the determinant can be easily calculated as the product of the diagonal elements. For finding the inverse, Gaussian elimination involves augmenting the original matrix with the identity matrix and performing row operations until the original matrix is reduced to the identity matrix, at which point the augmented portion becomes the inverse.
Gaussian elimination is generally more efficient for larger matrices and is better suited for computational methods. However, it may introduce numerical errors when dealing with floating-point arithmetic on computers. The cofactor expansion method, while less efficient for large matrices, does not suffer from these numerical issues and provides a clear algebraic formula for the determinant and inverse, which can be useful for theoretical analysis and proofs.
$$
where \(C_{ij}\) is the cofactor of element \(a_{ij}\), which is calculated as \(C_{ij} = (-1)^{i+j}M_{ij}\), and \(M_{ij}\) is the minor of \(a_{ij}\), the determinant of the submatrix that remains after removing the \(i\)-th row and \(j\)-th column from A.
The cofactors are:
$$
C_{11} = (-1)^{1+1}\text{det}\begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} = \text{det}\begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} = a_{22}a_{33} - a_{23}a_{32}
$$
$$
C_{12} = (-1)^{1+2}\text{det}\begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} = -\text{det}\begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} = -(a_{21}a_{33} - a_{23}a_{31})
$$
$$
C_{13} = (-1)^{1+3}\text{det}\begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} = \text{det}\begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} = a_{21}a_{32} - a_{22}a_{31}
$$
Substituting these cofactors into the determinant formula, we get:
$$
\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
$$
b) To prove that the determinant of matrix A can also be calculated by multiplying the elements of any row or column by their corresponding cofactors and adding the products, we will use the properties of determinants and cofactors. The determinant of a matrix is a linear function with respect to each row and column. This means that the determinant can be expressed as a sum of products of elements and their corresponding cofactors for any single row or column.
For any element \(a_{ij}\) in row \(i\), the determinant can be expressed as:
$$
\text{det}(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3}
$$
This is true for any row \(i\) and similarly for any column \(j\). The proof of this property is based on the multilinearity and alternating sign properties of the determinant function.
c) Given that the determinant of matrix A is equal to 5, we can use the formula derived in part (a) to find the value of \(a_{11}\) given the other elements:
$$
5 = a_{11}(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7)
$$
Solving for \(a_{11}\):
$$
5 = a_{11}(45 - 48) - 2(36 - 42) + 3(32 - 35)
$$
$$
5 = a_{11}(-3) + 12 - 9
$$
$$
5 = -3a_{11} + 3
$$
$$
2 = -3a_{11}
$$
$$
a_{11} = -\frac{2}{3}
$$
d) To find the inverse of matrix A, denoted as \(A^{-1}\), we use the formula:
$$
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
$$
where adj(A) is the adjugate of A, which is the transpose of the matrix of cofactors of A. Since we have already calculated the cofactors for the first row in part (a), we would need to calculate the cofactors for the remaining elements and then transpose the matrix of cofactors.
However, since the question does not provide all the necessary elements to calculate the full set of cofactors, we cannot complete this step without additional information. If all elements were known, we would calculate the cofactors for each element, form the matrix of cofactors, transpose it to get the adjugate, and then multiply by the reciprocal of the determinant to find the inverse.
e) The cofactor expansion method for finding the determinant and inverse of a 3x3 matrix is a direct method that involves calculating minors and cofactors. It is particularly useful for small matrices where the calculations are manageable. However, for larger matrices, this method becomes cumbersome due to the rapidly increasing number of minors that must be calculated.
In contrast, Gaussian elimination is an algorithmic approach that transforms the matrix into an upper triangular form, from which the determinant can be easily calculated as the product of the diagonal elements. For finding the inverse, Gaussian elimination involves augmenting the original matrix with the identity matrix and performing row operations until the original matrix is reduced to the identity matrix, at which point the augmented portion becomes the inverse.
Gaussian elimination is generally more efficient for larger matrices and is better suited for computational methods. However, it may introduce numerical errors when dealing with floating-point arithmetic on computers. The cofactor expansion method, while less efficient for large matrices, does not suffer from these numerical issues and provides a clear algebraic formula for the determinant and inverse, which can be useful for theoretical analysis and proofs.
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