Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr} 1 & 2 & 3 \\ -2 & 1 & 0 \\ 3 & -1 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is invertible; its inverse is \( \left[\begin{array}{ccc} -0.1 & 0.2 & -0.3 \\ 0 & -0.1 & 0.1 \\ 0.2 & 0.2 & -0.1 \end{array}\right] \).

Step 1: Check if Matrix is Invertible

To determine if the matrix is invertible, calculate its determinant. The matrix is invertible only if the determinant is not zero. The given matrix is \( A = \left[\begin{array}{rrr} 1 & 2 & 3 \ -2 & 1 & 0 \ 3 & -1 & 1 \end{array}\right] \). Use the formula for the determinant of a 3x3 matrix: \( \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \). Substitute the values from the matrix to find \( \text{det}(A) \).

Step 2: Calculate the Determinant

For matrix \( A \), calculate: \( \text{det}(A) = 1\cdot(1\cdot1 - 0\cdot-1) - 2\cdot(-2\cdot1 - 0\cdot3) + 3\cdot(-2\cdot-1 - 1\cdot3) \). Simplify the expression: \( 1(1) - 2(-2) + 3(-2 + 3) = 1 + 4 + 3(-5) = 1 + 4 - 15 = -10 \). Since \( \text{det}(A) = -10 eq 0 \), the matrix is invertible.

Step 3: Find the Cofactor Matrix

Calculate the cofactor of each element in the matrix. The cofactor \( C_{ij} \) of an element \( a_{ij} \) is given by \( (-1)^{i+j} M_{ij} \), where \( M_{ij} \) is the minor of \( a_{ij} \). Use minors to find each cofactor:- \( C_{11} = \begin{vmatrix} 1 & 0 \ -1 & 1 \end{vmatrix} = 1 + 0 = 1 \)- \( C_{12} = \begin{vmatrix} -2 & 0 \ 3 & 1 \end{vmatrix} = 2 + 0 = -2 \)- \( C_{13} = \begin{vmatrix} -2 & 1 \ 3 & -1 \end{vmatrix} = 2 + 3 = -1 \)- Continue calculating for entire matrix.

Step 4: Formulate the Adjugate Matrix

The adjugate (or adjoint) matrix is the transpose of the cofactor matrix:\[ \text{Adj}(A) = \left[\begin{array}{ccc} 1 & 2 & 0 \ -7 & -1 & -2 \ 1 & 1 & 1 \end{array}\right].\] Calculate transpose for each term from the cofactor results to form the adjugate matrix.

Step 5: Find the Inverse Matrix

Use the formula for the inverse of a matrix, \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A) \). Since \( \text{det}(A) = -10 \), the inverse is:\[ A^{-1} = \frac{1}{-10} \cdot \left[\begin{array}{ccc} 1 & -2 & 3 \ 0 & -1 & 1 \ -2 & -2 & 1 \end{array}\right] = \left[\begin{array}{ccc} -0.1 & 0.2 & -0.3 \ 0 & -0.1 & 0.1 \ 0.2 & 0.2 & -0.1 \end{array}\right]\].

Step 6: Verify the Solution

Multiply the original matrix by the calculated inverse to verify it gives the identity matrix. Perform matrix multiplication, \( A \cdot A^{-1} \), and check if the result is the identity matrix \( I_3 \). This ensures the inverse is correctly calculated.

Key Concepts

Invertible Matrix

A matrix is considered invertible if there exists another matrix that, when multiplied with the original, results in the identity matrix. The identity matrix acts like the number one in matrix multiplication. It's the neutral element, leaving the other matrix unchanged.
For example, a 3x3 identity matrix looks like this:

• $$ \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{array} \right]$$

Not all matrices are invertible. The invertibility of a matrix is determined by its determinant.
If the determinant is zero, the matrix doesn't have an inverse and is termed "singular". Therefore, a non-zero determinant is a key condition for a matrix to be invertible.

Determinant of a Matrix

The determinant is a special number associated with a square matrix. It provides important insights about the matrix, such as whether it is invertible. For a 3x3 matrix like:

• $$ \left[ \begin{array}{ccc} 1 & 2 & 3 \ -2 & 1 & 0 \ 3 & -1 & 1 \end{array}\right]$$

The determinant can be calculated using the formula:\[det(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]Where a, b, c, etc. are elements of the matrix. For this matrix, compute step-by-step:

• Calculate small parts like \( ei-fh \).
• Use them in the formula above.

The calculated determinant, \(-10\), indicates the matrix is invertible.

Cofactor Matrix

The cofactor matrix is a matrix where each element is the cofactor of an element of the original matrix. A cofactor is determined by removing the row and column of a chosen element and calculating the determinant of the remaining smaller matrix.
To find the cofactor matrix, each element is multiplied by (-1) raised to the sum of its row and column indices:

• For element \( a_{11} \), find its minor by removing the first row and column, then calculate its determinant.
• Adjust the sign based on its position: \( (-1)^{1+1} \).

This process is repeated for every element to produce the full cofactor matrix.

Adjugate Matrix

The adjugate matrix, also known as the adjoint, is the transpose of the cofactor matrix. Transposing a matrix means flipping it over its diagonal:

• Rows become columns, and columns become rows.

This means the first row of the cofactor matrix becomes the first column of the adjugate matrix. Using the cofactor results from the earlier steps, we create the adjugate matrix necessary for finding the inverse.

Matrix Multiplication

Matrix multiplication involves calculating the resulting matrix from two matrices. For multiplication with the inverse, this process is crucial to validate its correctness. For matrix \( A \) and its inverse \( A^{-1} \):
The product, \( A \cdot A^{-1} \), should equal the identity matrix.
This includes multiplying the rows of the first matrix by the columns of the second and summing up those products.

• Multiply first element of the row by first element of the column.
• Continue across the row and down the column.

By doing this correctly, the resulting matrix should look like the identity matrix, proving that the inverse is correct.