Problem 24 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{lll} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse matrix \(A^{-1}\) is: \[ \begin{bmatrix} \frac{7}{4} & -\frac{1}{4} & -\frac{1}{4} \\ -\frac{3}{4} & \frac{1}{4} & 0 \\ -\frac{3}{4} & 0 & \frac{1}{4} \end{bmatrix} \]

Step 1: Understand the Inverse of a Matrix

To find the inverse of a matrix, we need to ensure the matrix is square and has a non-zero determinant. The inverse exists only for non-singular matrices.

Step 2: Determine if the Matrix is Square

Matrix \(A\) is a 3x3 matrix (3 rows and 3 columns), so it's a square matrix. Thus, it is possible for an inverse to exist.

Step 3: Calculate the Determinant of Matrix A

We calculate the determinant of \(A\) using the formula for a 3x3 matrix: \[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Substituting the elements of matrix \(A\): \[\text{det}(A) = 1(4 \cdot 4 - 3 \cdot 3) - 3(1 \cdot 4 - 3 \cdot 1) + 3(1 \cdot 3 - 3 \cdot 1)\]Simplifying gives:\[= 1(16 - 9) - 3(4 - 3) + 3(3 - 3)\]\[= 1 \cdot 7 - 3 \cdot 1 + 3 \cdot 0\]\[= 7 - 3 + 0 = 4\] Since the determinant is 4, which is not zero, the inverse of matrix \(A\) exists.

Step 4: Use the Formula for the Inverse of a 3x3 Matrix

The inverse of a 3x3 matrix \(A\), if it exists, is computed as:\[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\]First, we'll find the adjoint (adjugate) matrix \(\text{adj}(A)\) by finding the cofactor matrix and then transposing it.

Step 5: Compute the Cofactor Matrix

For each element \(a_{ij}\), compute the cofactor by eliminating the i-th row and j-th column and calculating the 2x2 determinant of the remaining matrix, then applying the sign \((-1)^{i+j}\).For example, the first cofactor \(C_{11}\) is \[C_{11} = +\begin{vmatrix} 4 & 3 \ 3 & 4 \end{vmatrix} = (4 \cdot 4 - 3 \cdot 3) = 7\]Follow this method for all elements. The cofactor matrix becomes:\[\begin{bmatrix}7 & -3 & -3 \ -1 & 1 & 0 \ -1 & 0 & 1\end{bmatrix}\]

Step 6: Transpose the Cofactor Matrix to Find the Adjoint

The transpose of the cofactor matrix, gives us the adjoint matrix:\[\text{adj}(A) = \begin{bmatrix}7 & -1 & -1 \-3 & 1 & 0 \-3 & 0 & 1\end{bmatrix}\]

Step 7: Calculate the Inverse Matrix

Finally, use the formula:\[A^{-1} = \frac{1}{4} \cdot \begin{bmatrix}7 & -1 & -1 \-3 & 1 & 0 \-3 & 0 & 1\end{bmatrix}\]Thus, \[A^{-1} = \begin{bmatrix}\frac{7}{4} & -\frac{1}{4} & -\frac{1}{4} \-\frac{3}{4} & \frac{1}{4} & 0 \-\frac{3}{4} & 0 & \frac{1}{4}\end{bmatrix}\]

Key Concepts

Determinant of a Matrix

The determinant is a crucial value associated with a square matrix. It condenses the matrix into a single number that reveals several properties.
Specifically, it tells us whether the matrix is invertible. For a given 3x3 matrix, the determinant is calculated as follows:

• Pick an element from the first row and multiply it by the determinant of the 2x2 matrix leftover after removing that element's row and column.
• Apply the same process to the rest of the elements in the first row, alternating the signs (positive, negative, positive, etc.).

In the example exercise, the matrix determinant is computed using the formula \[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\], resulting in a value of 4, meaning the matrix is invertible because its determinant is non-zero.
Understanding how to calculate the determinant helps in determining whether an inverse matrix can exist.

Cofactor Matrix

The cofactor matrix is a key component in finding the inverse of a matrix. This matrix consists of the cofactors of each element of the original matrix.
Each cofactor is calculated by:

• Removing the row and column of the element we're examining.
• Calculating the determinant of the resulting 2x2 matrix.
• Multiplying this determinant by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices, respectively.

For instance, in our exercise, the cofactor of the element in position (1,1) was calculated by taking the determinant of the leftover 2x2 matrix \[\begin{vmatrix} 4 & 3 \ 3 & 4 \end{vmatrix}\] and multiplying it by \(1\) (since \((-1)^{1+1} = 1\)).
The cofactor matrix for our example becomes \[\begin{bmatrix} 7 & -3 & -3 \ -1 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix}\]. This step is essential for finding the adjugate matrix.

Adjugate Matrix

The adjugate matrix, also known as the adjoint, is derived from the cofactor matrix and is crucial for determining the inverse of a square matrix.
To obtain the adjugate matrix:

• First compute the cofactor matrix as previously discussed.
• Then, take the transpose of the cofactor matrix, swapping rows with columns.

In the provided example, the cofactor matrix \[\begin{bmatrix} 7 & -3 & -3 \ -1 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix}\] is transposed to get the adjugate matrix \[\begin{bmatrix} 7 & -1 & -1 \ -3 & 1 & 0 \ -3 & 0 & 1 \end{bmatrix}\].
This step helps form the inverse of a matrix by properly adjusting the elements according to their positions.

Square Matrix

A square matrix has an equal number of rows and columns. It is a primary requirement for determining an inverse because non-square matrices lack some of the necessary properties.
Only square matrices can have determinants and, thus, the possibility of inverses. A non-square matrix, regardless of its size, does not meet the requirements for invertibility.
For example, matrix \(A\) in the exercise is a 3x3 square matrix, meaning it has the potential to be inverted. This attribute is what allowed us to explore finding its inverse by using its determinant, cofactor matrix, and adjugate matrix.
The concept of a square matrix extends beyond invertibility, being key in various matrix operations, including computing eigenvalues, powers of matrices, and solving systems of linear equations.