Problem 58 - Precalculus

Question

find \(A^{-1},\) if possible. $$A=\left[\begin{array}{rrr}1 & 2 & -3 \\\1 & -1 & -1 \\\1 & 0 & -4\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix}4 & 1 & 1 \\ 3 & -1 & 4 \\ 1 & 0 & -1\end{bmatrix} \).

Step 1: Check if the Matrix is Square

To find the inverse of a matrix, it must be square. A square matrix has the same number of rows and columns. Matrix \( A \) is a 3x3 matrix, so it is square, and we can proceed to find its inverse.

Step 2: Find the Determinant of the Matrix

The matrix must have a non-zero determinant to be invertible. Calculate the determinant of matrix \( A \) using the formula for a 3x3 matrix:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) = 1((-1)(-4)-0(-1)) - 2(1(-4)-(-1)(1)) + (-3)(1(0)-(-1)(1))\]Simplify to find:\[\text{det}(A) = 4 - 2(3) + 3 = 4 - 6 + 3 = 1\] The determinant is 1, which is non-zero. Thus, an inverse exists.

Step 3: Compute the Adjugate (Adjoint) of the Matrix

The adjugate of a matrix is the transpose of the cofactor matrix. First, find the cofactor matrix for \( A \):- Cofactor of \(a_{11} = (-1)^1 \times (\text{det}(\begin{vmatrix}-1 & -1 \ 0 & -4\end{vmatrix})) = 4\)- Cofactor of \(a_{12} = (-1)^2 \times (\text{det}(\begin{vmatrix}1 & -1 \ 1 & -4\end{vmatrix})) = -3\)- Cofactor of \(a_{13} = (-1)^3 \times (\text{det}(\begin{vmatrix}1 & -1\ 1 & 0\end{vmatrix})) = 1\)- Repeat for other entries similarly. Thus, the cofactor matrix is:\[\begin{bmatrix}4 & 3 & 1 \1 & -1 & 0 \1 & 4 & -1\end{bmatrix}\]Transpose this matrix to get the adjugate matrix:\[\begin{bmatrix}4 & 1 & 1 \3 & -1 & 4 \1 & 0 & -1\end{bmatrix}\]

Step 4: Calculate the Inverse Using the Determinant and Adjugate

The inverse of matrix \( A \) can be computed using the formula \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \). Since the determinant is 1, we have:\[A^{-1} = 1 \cdot \begin{bmatrix}4 & 1 & 1 \3 & -1 & 4 \1 & 0 & -1\end{bmatrix}\]Thus, the inverse matrix is:\[\begin{bmatrix}4 & 1 & 1 \3 & -1 & 4 \1 & 0 & -1\end{bmatrix}\]

Step 5: Verify the Inverse (Optional)

To confirm the correctness of \( A^{-1} \), multiply \( A \times A^{-1} \) and check if it equals the identity matrix. Compute \( A \times A^{-1} \):\[\begin{bmatrix}1 & 2 & -3 \ 1 & -1 & -1 \ 1 & 0 & -4\end{bmatrix} \begin{bmatrix}4 & 1 & 1 \ 3 & -1 & 4 \ 1 & 0 & -1\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}\]The product is the identity matrix, confirming that our computed inverse is correct.

Key Concepts

Determinant Calculation

The determinant of a matrix plays a crucial role in determining whether a matrix can have an inverse. A non-zero determinant indicates that an inverse exists. Finding the determinant of a 3x3 matrix involves a specific formula that uses minors and co-factors. For a matrix \( A \) described by

• \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \)

the determinant is calculated as:

• \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \)

In our example, substituting the values from matrix \( A \) gives us:

• \( \text{det}(A) = 1((-1)(-4) - 0(-1)) - 2(1(-4) - (-1)(1)) + (-3)(1(0) - (-1)(1)) \)
• This simplifies to: \( 4 - 6 + 3 = 1 \)

The determinant in this instance is 1, a non-zero value, which confirms the matrix is invertible.

Adjugate Matrix

The adjugate matrix is used in computing the inverse of a matrix and is derived from the cofactor matrix. It involves transposing the cofactor matrix. For any square matrix, the adjugate can be defined simply as the transpose of its cofactor matrix.To compute the adjugate matrix, first, find the cofactor matrix, then transpose it. If matrix \( A \) is given and the cofactor matrix is found as

• \( \begin{bmatrix} 4 & 3 & 1 \ 1 & -1 & 0 \ 1 & 4 & -1 \end{bmatrix} \)

then the adjugate of \( A \) is the transposed version:

• \( \text{adj}(A) = \begin{bmatrix} 4 & 1 & 1 \ 3 & -1 & 4 \ 1 & 0 & -1 \end{bmatrix} \)

Transposing involves flipping the matrix over its diagonal, swapping the row and column indices of each element.

Cofactor Matrix

The cofactor matrix includes values that are critical for determining the adjugate matrix, which ultimately helps in finding the inverse. The cofactor of an element in a matrix is essentially its minor with a sign change based on its position.

• The cofactor of an element \( a_{ij} \) is given as \( (-1)^{i+j} M_{ij} \), where \( M_{ij} \) is the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column of \( A \).

Taking each element from matrix \( A = \begin{bmatrix} 1 & 2 & -3 \ 1 & -1 & -1 \ 1 & 0 & -4 \end{bmatrix} \), calculate the cofactors:

• The cofactor of \( a_{11} = 4 \), calculated as \( (-1)^2 \times \text{det}(\begin{vmatrix} -1 & -1 \ 0 & -4 \end{vmatrix}) \)
• The cofactor of \( a_{12} = -3 \), calculated similarly for its submatrix, and so on.
• Create a cofactor matrix \( \begin{bmatrix} 4 & 3 & 1 \ 1 & -1 & 0 \ 1 & 4 & -1 \end{bmatrix} \)

This cofactor matrix needs to be transposed to find the adjugate matrix, which is essential for inverting the matrix.