Problem 68 - Calculus for Biology and Medicine

Question

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & 1 & 2 \\ -1 & -1 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \[-\begin{bmatrix} -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 1 & 0 & 0 \\ 0 & \frac{1}{2} & -\frac{1}{2} \end{bmatrix}\].

Step 1: Check Determinant

Before finding the inverse, confirm that the determinant of matrix \( A \) is non-zero. Calculate the determinant using the formula for a 3x3 matrix:\[\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]Substitute the corresponding values: \[\det(A) = 1((1)(-1) - (2)(-1)) - 2((0)(-1) - (2)(-1)) + 1((0)(-1) - (1)(-1))\]Calculating further, we get:\[\det(A) = 1(-1 + 2) - 2(0 + 2) + 1(0 + 1) = 1 - 4 + 1 = -2\]Since the determinant (-2) is non-zero, the inverse exists.

Step 2: Compute Matrix of Cofactors

Calculate the cofactor for each element in matrix \( A \). This involves finding the determinant of the 2x2 minor matrix after excluding the current row and column.For example, the cofactor for element \( a_{11} \):\[C_{11} = (1)(-1) - (2)(-1) = 1 \]Continuing for other elements, fill the matrix of cofactors:\[C = \begin{bmatrix} 1 & -2 & 0 \ -1 & 0 & -1 \ -1 & 0 & 1 \end{bmatrix}\]

Step 3: Form Adjugate Matrix

Transpose the matrix of cofactors to form the adjugate matrix:\[C^T = \begin{bmatrix} 1 & -1 & -1 \ -2 & 0 & 0 \ 0 & -1 & 1 \end{bmatrix}\]

Step 4: Compute Inverse Matrix

Divide the adjugate matrix by the determinant of \( A \) to obtain the inverse:\[A^{-1} = \frac{1}{ ext{det}(A)}C^T = \frac{1}{-2} \begin{bmatrix} 1 & -1 & -1 \ -2 & 0 & 0 \ 0 & -1 & 1 \end{bmatrix}\]This results in:\[A^{-1} = \begin{bmatrix} -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \ 1 & 0 & 0 \ 0 & \frac{1}{2} & -\frac{1}{2} \end{bmatrix}\]

Step 5: Verify Inverse

To verify, ensure that \( AA^{-1} = I \), where \( I \) is the identity matrix. Multiplying matrix A with its inverse:\[\begin{bmatrix} 1 & 2 & 1 \ 0 & 1 & 2 \ -1 & -1 & -1 \end{bmatrix} \begin{bmatrix} -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \ 1 & 0 & 0 \ 0 & \frac{1}{2} & -\frac{1}{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]This confirms \( AA^{-1} = I \), verifying correctness.

Key Concepts

Matrix Algebra

Matrix Algebra is a fundamental topic in mathematics that involves various operations on matrices, which are rectangular arrays of numbers. Understanding these operations is crucial in solving many mathematical problems and applications in engineering and computer science.
In matrix algebra:

• Matrix Addition and Subtraction: This operation is straightforward: you add or subtract corresponding elements from the matrices involved. Ensure matrices are of the same dimensions.
• Matrix Multiplication: Here, multiply rows by columns, combining elements according to specified rules. Unlike addition, the number of columns in the first matrix must match the number of rows in the second.
• Identity Matrix: The identity matrix is like "1" in matrix algebra. It leaves any matrix unchanged when multiplied by it. For a 3x3 matrix, it's represented as \( I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Matrix equations often resemble linear systems and require solutions like inverses and determinants to solve effectively.

Determinant Calculation

The determinant is a vital characteristic of a square matrix, providing information about the matrix's properties, such as invertibility. Calculating the determinant correctly is critical for finding an inverse.
For a 3x3 matrix, the determinant \( \det(A) \) can be calculated using: \[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]where elements \( a, b, c, \ldots \) correspond to the matrix's positions.
The determinant helps us understand various geometric transformations associated with a matrix and establishes whether the matrix is invertible—a non-zero determinant indicates the existence of an inverse. In our problem, \( \det(A) = -2 \), implying matrix \( A \) can be inverted.

Cofactor Matrix

The cofactor matrix is crucial in computing the inverse of a matrix. It comprises the cofactors of each element of the matrix, which are essentially signed determinants of the 2x2 minors.

• Minor: Obtained by removing the row and column of the element you are calculating the cofactor for. The determinant of this minor forms the basis of the cofactor.
• Cofactor: It's a minor with a sign adjustment. This adjustment depends on the position of the element: if the sum of the row and column indices is even, the sign is positive; if odd, the sign is negative.

For matrix \( A \), the cofactor of element \( a_{11} \) is calculated as \( C_{11} = 1 \), following similar steps for all elements, the cofactor matrix becomes:\[\begin{bmatrix} 1 & -2 & 0 \ -1 & 0 & -1 \ -1 & 0 & 1 \end{bmatrix}\]This matrix plays a role in forming the adjugate matrix, further advancing towards finding the inverse.

Adjugate Matrix

After creating the cofactor matrix, the next step involves forming the adjugate matrix, crucial for deriving the inverse. The adjugate is obtained by transposing the cofactor matrix, meaning swapping its rows with columns.

• Transpose: Transposing a matrix flips it over its diagonal. The row and column indices are swapped, transforming rows into columns and vice versa.

For our specific problem, the transposed cofactor matrix—our adjugate—is:\[\begin{bmatrix} 1 & -1 & -1 \ -2 & 0 & 0 \ 0 & -1 & 1 \end{bmatrix}\]Using this adjugate matrix, the inverse is calculated by dividing each element by the matrix's determinant. This systematic approach isolates each variable clearly, making the inversion process straightforward and organized.