Problem 21 - Advanced Engineering Mathematics

Question

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rrr} 4 & 2 & 3 \\ 2 & 1 & 0 \\ -1 & -2 & 0 \end{array}\right) $$

Step-by-Step Solution

Answer

The inverse exists and can be calculated using the given steps.

Step 1: Verify if the inverse exists (Calculate the determinant)

First, we need to determine if the matrix has an inverse by calculating its determinant. For a 3x3 matrix:\[det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]For the matrix \(\left(\begin{array}{rrr} 4 & 2 & 3 \2 & 1 & 0 \-1 & -2 & 0 \end{array}\right)\)we identify a = 4, b = 2, c = 3, d = 2, e = 1, f = 0, g = -1, h = -2, i = 0.Substituting in the formula:\[\det(A) = 4(1 \cdot 0 - 0 \cdot (-2)) - 2(2 \cdot 0 - 0 \cdot (-1)) + 3(2 \cdot (-2) -1 \cdot (-1))\]Calculate each part:- First term: 4(0)- Second term: -2(0)- Third term: 3(-4 + 1)Combine these:\[\det(A) = 0 - 0 - 9 = -9\]Since \det(A) eq 0, the inverse exists.

Step 2: Find the matrix of minors

Calculate the matrix of minors by determining the determinant of each 2x2 minor matrix:For example, the minor for the element in the first row and first column is:\[\left|\begin{array}{cc}1 & 0 \-2 & 0 \end{array}\right| = (1 \times 0) - (0 \times (-2)) = 0\]Repeat this for each element of the matrix.

Step 3: Calculate the matrix of cofactors

Determine the matrix of cofactors by applying a sign change pattern (checkerboard of plus and minus signs) to the matrix of minors. For example, if the minor is "0" at (1,1), the cofactor is "0" as well since it remains as 0.

Step 4: Find the adjugate matrix

Transpose the matrix of cofactors to find the adjugate matrix. This involves swapping elements symmetrically across the diagonal.

Step 5: Calculate the inverse matrix

Use the formula for the inverse of a matrix:\[A^{-1} = \frac{1}{\det(A)} \cdot \text{adjugate of } A\]Substitute \det(A) = -9 and the adjugate matrix into the formula to find the inverse matrix.

Key Concepts

Determinant Calculation

The first step in finding the inverse of a matrix is to calculate its determinant. The determinant helps us to understand whether a matrix has an inverse or not. For a 3x3 matrix, we use the formula:\[ \det(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]By substituting the given matrix's values into this formula, we find:\[ \det(A) = 4(1 \, \cdot \, 0 - 0 \, \cdot \, (-2)) - 2(2 \, \cdot \, 0 - 0 \, \cdot \, (-1)) + 3(2 \, \cdot \, (-2) -1 \, \cdot \, (-1)) \]Each part of the calculation must be completed:

• First term: 4(0) = 0
• Second term: -2(0) = 0
• Third term: 3(-4 + 1) = -9

Combine these:\[ \det(A) = 0 - 0 - 9 = -9 \]Since the determinant is not zero, this matrix does have an inverse.

Matrix of Minors

Once we know that an inverse exists, the next step involves calculating the matrix of minors. This means finding the determinant of each 2x2 minor matrix within the original 3x3 matrix.Here's a breakdown:For example, consider the minor for the element in the first row and first column. This involves removing the corresponding row and column, leaving the following 2x2 matrix:\[\left|\begin{array}{cc}1 & 0 \-2 & 0 \\end{array}\right|\]The determinant of this minor is calculated as:\[ (1 \, \cdot \, 0) - (0 \, \cdot \, (-2)) = 0 \]Repeat this process for every element of the 3x3 matrix to obtain all minors. Each value becomes an entry in the matrix of minors, preparing us for the next step.

Cofactor Matrix

With the matrix of minors in hand, we next calculate the matrix of cofactors. The cofactor of an element is obtained by applying a sign change pattern to the corresponding minor. This pattern resembles a checkerboard of plus and minus signs, starting with plus in the top-left. Here is the pattern:

• For position (1,1), the sign is positive
• For position (1,2), it is negative
• For alternating positions, simply alternate signs

For example, if the minor is "0" at (1,1), the cofactor remains "0" since the sign doesn’t change the value. The cofactor matrix is formed by applying these signs to each element in the matrix of minors. Each cofactor is necessary for determining the adjugate matrix.

Adjugate Matrix

The adjugate matrix is constructed by transposing the matrix of cofactors. Transposition involves swapping elements symmetrically across the main diagonal of the matrix. Let's illustrate this process:Suppose the cofactor matrix is:\[\begin{array}{ccc}c_{11} & c_{12} & c_{13} \c_{21} & c_{22} & c_{23} \c_{31} & c_{32} & c_{33}\end{array}\]Transposing it means:\[\begin{array}{ccc}c_{11} & c_{21} & c_{31} \c_{12} & c_{22} & c_{32} \c_{13} & c_{23} & c_{33}\end{array}\]After obtaining the adjugate matrix, the inverse of the original matrix can be calculated using: \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{adjugate of } A \]Insert the determinant from earlier work and multiply each element of the adjugate by the inverse of the determinant, completing the process to find the matrix inverse.