Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{ccc} 1 & 0 & 6 \\ -2 & 1 & 7 \\ 3 & 0 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix has an inverse: \( \begin{bmatrix} -\frac{1}{16} & \frac{14}{16} & \frac{1}{8} \\ -\frac{3}{16} & \frac{1}{4} & -\frac{3}{8} \\ \frac{3}{8} & -\frac{3}{4} & -\frac{1}{16} \end{bmatrix} \).

Step 1: Check if the matrix is square

The given matrix is \(3 \times 3\), meaning it is a square matrix. Only square matrices can have inverses.

Step 2: Calculate the determinant of the matrix

For the matrix \( A = \begin{bmatrix} 1 & 0 & 6 \ -2 & 1 & 7 \ 3 & 0 & 2 \end{bmatrix} \), the determinant \( \text{det}(A) \) is calculated as follows: \[ \text{det}(A) = 1(1 \cdot 2 - 7 \cdot 0) - 0(-2 \cdot 2 - 7 \cdot 3) + 6(-2 \cdot 0 - 1 \cdot 3) \] Simplify the expression:\[ \text{det}(A) = 1(2) + 0 + 6(-3) = 2 - 18 = -16 \]

Step 3: Determine if the inverse exists

Since the determinant \( \text{det}(A) = -16 \) is non-zero, the matrix has an inverse. If the determinant were zero, the matrix would not have an inverse.

Step 4: Calculate the adjoint matrix

The adjoint of a \(3 \times 3\) matrix \(A\) is the transpose of its cofactor matrix. Calculate each minor, find the cofactors, and then transpose to find the adjoint. The cofactor matrix is:\[\begin{bmatrix}1 & 3 & -6 \-14 & -4 & 12 \-2 & 6 & 1\end{bmatrix}\]Transpose the cofactor matrix to get the adjoint:\[\text{adj}(A) = \begin{bmatrix}1 & -14 & -2 \3 & -4 & 6 \-6 & 12 & 1\end{bmatrix}\]

Step 5: Compute the inverse of the matrix

The inverse of matrix \( A \) is given by \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \). Since \( \text{det}(A) = -16 \), we have:\[ A^{-1} = \frac{1}{-16} \begin{bmatrix} 1 & -14 & -2 \ 3 & -4 & 6 \ -6 & 12 & 1 \end{bmatrix} = \begin{bmatrix} -\frac{1}{16} & \frac{14}{16} & \frac{1}{8} \ -\frac{3}{16} & \frac{1}{4} & -\frac{3}{8} \ \frac{3}{8} & -\frac{3}{4} & -\frac{1}{16} \end{bmatrix} \]

Key Concepts

Square Matrix

When dealing with matrix inversion, the concept of a square matrix is fundamental. A square matrix is simply a matrix that has the same number of rows and columns. This regularity is crucial because only square matrices can potentially have a multiplicative inverse.

• A matrix with dimensions such as 3x3, like the one given in the exercise, is a square matrix.
• Non-square matrices lack this uniformity, preventing any possibility of finding an inverse.

In the context of the exercise, confirming that our matrix is 3x3 is a crucial initial step before checking for its inverse.

Determinant Calculation

The determinant is a unique number that can be calculated for square matrices. This number is vital for determining whether a matrix has an inverse. The determinant calculation for a 3x3 matrix may seem intricate, but breaking it down makes it feasible.

• For our example, the determinant is calculated using:\[ \text{det}(A) = 1(1 \cdot 2 - 7 \cdot 0) - 0(-2 \cdot 2 - 7 \cdot 3) + 6(-2 \cdot 0 - 1 \cdot 3) \]
• This simplifies to \[ \text{det}(A) = 2 - 18 = -16 \].

A non-zero determinant, like -16 in this exercise, indicates that an inverse exists. Conversely, a zero determinant means the matrix is singular and lacks an inverse.

Adjoint of a Matrix

The adjoint of a matrix plays a crucial role in finding the inverse of a matrix. It is essentially the transpose of the cofactor matrix. This requires calculation in two steps: finding cofactors and transposing the matrix.

• First, calculate the cofactors for each element of the matrix.
• Next, take the transpose of the cofactor matrix to get the adjoint.

For instance, in our exercise, the cofactor matrix is:\[ \begin{bmatrix} 1 & 3 & -6 \ -14 & -4 & 12 \ -2 & 6 & 1 \end{bmatrix} \]which transposes to give the adjoint:\[ \text{adj}(A) = \begin{bmatrix} 1 & -14 & -2 \ 3 & -4 & 6 \ -6 & 12 & 1 \end{bmatrix} \].This transformed version of the matrix is essential for computing the inverse.

Matrix Cofactors

Understanding matrix cofactors is pivotal when finding the adjoint and subsequently the inverse of a matrix. Cofactors are specific minors of the matrix, essentially determinants of smaller square matrices derived from the larger matrix.

• To find a cofactor for an element, remove its row and column and compute the determinant of the resulting submatrix.
• The cofactor is then adjusted by a positive or negative sign based on its position.

This process iteratively applied helps in constructing the full cofactor matrix, which in this exercise is:\[ \begin{bmatrix} 1 & 3 & -6 \ -14 & -4 & 12 \ -2 & 6 & 1 \end{bmatrix} \].Calculating cofactors is indispensable as they serve as the building blocks for the adjoint matrix.