Problem 50 - Differential Equations and Linear Algebra

Question

Find the specified element in the inverse of the given matrix. Do not use elementary row operations. \(A=\left[\begin{array}{rrr}-1 & -2 & 4 \\ 0 & 2 & -1 \\ 3 & -2 & 1\end{array}\right] ; \quad(1,1)-\) clement.

Step-by-Step Solution

Answer

The (1, 1) element of the inverse of the given matrix A is 0.

Step 1: Calculate the determinant of matrix A

To find the determinant of the given 3x3 matrix A, we will use the formula: \[\text{det}(A)= a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\] where \(a_{ij}\) represents the elements of matrix A. Using this formula, we have: \[\text{det}(A) = (-1)((2)(1) - (-1)(-2)) - (-2)((0)(1) - (-1)(3)) + (4)((0)(-2) - (2)(3))\]

Step 2: Simplify the determinant expression

Simplifying the previous expression, we get: \[\text{det}(A) = (-1)(2 - 2) + (2)(0 - (-3)) + (4)(0 - (-6))\] \[\text{det}(A) = 0 + 6 + 24 = 30\]

Step 3: Find the adjoint of matrix A

The adjoint of a matrix is the transpose of the matrix of its cofactors. To find the adjoint, we will calculate the cofactors for each element and then transpose the result: \(\text{adj}(A)= \left[\begin{array}{ccc} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33}\end{array}\right]\) where \(C_{ij}\) represents the cofactor of the corresponding element. Calculating the cofactors, we get: \(C_{11} = (2)(1) - (-1)(-2) = 0\) \(C_{12} = -((0)(1) - (-1)(3)) = -3\) \(C_{13} = (0)(-2) - (2)(3) = -6\) \(C_{21} = -((-2)(1) - (4)(-1)) = 6\) \(C_{22} = -((-1)(1) - (4)(3)) = 11\) \(C_{23} = -((-1)(-2) - (2)(3)) = -8\) \(C_{31} = (2)(-1) -((-1)(-2)) = -4\) \(C_{32} = -((0)(-1) - (4)(3)) = -12\) \(C_{33} = (0)(-2) - ((-2)(3)) = 6\) So, the adjoint of A becomes: \(\text{adj}(A)= \left[\begin{array}{ccc} 0 & -3 & -6 \\\ 6 & 11 & -8 \\\ -4 & -12 & 6\end{array}\right]\)

Step 4: Find the inverse of matrix A

The inverse of matrix A, denoted as \(A^{-1}\), is given by the formula: \[A^{-1} = \frac{1}{\text{det}(A)} * \text{adj}(A)\] Using the determinant calculated in step 2 and the adjoint from step 3, we get: \(A^{-1} = \frac{1}{30} * \left[\begin{array}{ccc} 0 & -3 & -6 \\\ 6 & 11 & -8 \\\ -4 & -12 & 6\end{array}\right]\)

Step 5: Find the (1, 1) element of the inverse

Now, to find the (1, 1) element of the inverse of matrix A, we look at the (1, 1) position in the resulting matrix from the previous step: \((A^{-1})_{11} = \frac{1}{30} * (0) = 0\) Thus, the (1, 1) element of the inverse of matrix A is 0.

Key Concepts

Determinant Calculation

Calculating the determinant of a matrix is a crucial skill in linear algebra. For a 3x3 matrix, like the one given, we use a specific formula. The determinant of matrix \(A\) is calculated as follows:

• Multiply each element of the first row by its corresponding cofactor.
• Use the formula: \[\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]
• Substitute the elements to simplify the expression.

This results in a single numeric value that is indicative of certain matrix properties, like invertibility.

Cofactor Method

The cofactor method is used to determine the adjoint of a matrix. Each element of a matrix has a cofactor, which is vital in matrix transformations.

• The cofactor \(C_{ij}\) of an element \(a_{ij}\) is obtained by removing the \(i\)th row and \(j\)th column and calculating the determinant of the resulting minor matrix.
• For the given matrix, we calculated several cofactors, such as:\[ C_{11} = (2)(1) - (-1)(-2) = 0 \]
• Remember to apply a sign change pattern, \((-1)^{i+j}\), to each cofactor.

The collection of cofactors forms the cofactor matrix.

Adjoint of a Matrix

The adjoint, or adjugate, of a matrix is an essential step towards finding an inverse.

• The adjoint is the transpose of the cofactor matrix.
• Once we have the cofactor matrix:\[\text{adj}(A)= \left[\begin{array}{ccc}0 & -3 & -6 \ 6 & 11 & -8 \ -4 & -12 & 6\end{array}\right]\]
• Transpose it by swapping rows and columns.

The adjoint aids in finding the inverse when the determinant is non-zero.

Matrix Notation

Matrix notation provides a compact way to represent and solve systems of linear equations.

• A matrix is typically denoted by a capital letter, such as \(A\).
• Elements within a matrix are represented by a lower-case letter with indices, such as \(a_{ij}\), where \(i\) and \(j\) indicate the row and column.
• This notation helps in organizing and performing operations systematically.

Using these notations makes complex operations like determinations and inverses more accessible for solving algebraic problems.