Problem 17 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

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

Step 1: Verify if the Matrix is Square

The first step in finding the inverse of a matrix is to check if it is square, which means it should have the same number of rows and columns. The given matrix \(A\) is a 3x3 matrix. Therefore, it is square.

Step 2: Calculate the Determinant

For a 3x3 matrix, the determinant \( \text{det}(A) \) needs to be non-zero for an inverse to exist. Calculate the determinant using the formula:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]For matrix \( A = \begin{bmatrix} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix} \), substitute:\[\text{det}(A) = 0(0 \cdot 0 - 1 \cdot 1) - 0(1 \cdot 0 - 0 \cdot 0) + 1(1 \cdot 1 - 0 \cdot 0) = 1\]Since the determinant is 1, the inverse exists.

Step 3: Find the Adjoint Matrix

The adjoint of a 3x3 matrix is the transpose of its cofactor matrix. Find the cofactor for each element and form the cofactor matrix. For matrix \( A \):- The cofactor of element at (1,1): \(+1(0 \cdot 0 - 1 \cdot 0) = 0\)- The cofactor of element at (1,2): \(-1(1 \cdot 0 - 0 \cdot 0) = 0\)- The cofactor of element at (1,3): \(+1(1 \cdot 1 - 0 \cdot 0) = 1\)Continue this for all elements and transpose the cofactor matrix:\[\text{Adj}(A) = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 0 \end{bmatrix}\]

Step 4: Compute the Inverse Using Adjoint

The inverse of matrix \( A \) is given by the formula:\[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A)\]Substitute the known values:\[A^{-1} = \frac{1}{1} \cdot \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 0 \end{bmatrix}\]

Key Concepts

Determinant

The determinant of a matrix is a special number that can be calculated from its elements. It's an important factor in finding out if a matrix can be inverted. For a square matrix, if the determinant is zero, it means the matrix doesn't have an inverse.
To calculate the determinant of a 3x3 matrix, like matrix \( A \) in our exercise, use the specific formula:

• Identify elements of the matrix as \( a, b, c \) in the first row, \( d, e, f \) in the second row, and \( g, h, i \) in the third row.
• Apply the formula: \( ext{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\).

In our case, the calculation returned a determinant of 1, which is non-zero, confirming that an inverse does exist for the matrix \( A \). This step is crucial because it assures us that we can move forward to find other components needed for the inverse.

Adjoint Matrix

The adjoint matrix, also known as the adjugate matrix, is a helpful tool when calculating the inverse of a square matrix.
To find the adjoint:

• Calculate the cofactor for each entry in the matrix and form a new matrix from these cofactors. This matrix is known as the cofactor matrix.
• Transpose this cofactor matrix. Transposing involves flipping the matrix over its diagonal, swapping rows with columns.

The adjoint of matrix \( A \) was found by transposing its cofactor matrix. This adjoint matrix played a key role in finding \( A^{-1} \). When you understand how to find the adjoint, solving for the inverse becomes much smoother.

Cofactor Matrix

The cofactor matrix is an essential part of calculating the adjoint. Each element in a cofactor matrix is a signed minor of the original matrix. Let’s break it down:

• The "minor" of an element is the determinant of the submatrix that remains after removing the row and column of that element.
• The "sign" follows a checkerboard pattern starting with a positive at the top left, alternating signs in a consistent manner.

For each element in matrix \( A \), you find its cofactor, which involves calculating its minor and applying the correct sign. In matrix \( A \), the cofactor matrix was calculated, and later it became transposed to form the adjoint. Understanding cofactors is like having the building blocks for creating an adjoint matrix.

Square Matrix

A square matrix is simply a matrix with the same number of rows and columns. This property is a prerequisite for calculating an inverse.

• Identifying if a matrix is square is the first step.
• Only square matrices can potentially have an inverse.

In our exercise, matrix \( A \) was a 3x3 square matrix, which is essential because only then could we proceed to find its determinant and adjoint. Without being square, the calculations for inverses we performed wouldn't be valid. Recognizing a square matrix ensures you're looking at the right kind of matrix for inversion purposes.